Booklet for the 30Th Chemnitz FEM Symposium

Chemnitz FEM-Symposium 2017

Programme

Collection of abstracts List of participants

St. Wolfgang/Strobl, September 25 - 27, 2017 Scientiﬁc Topics:

The symposium is devoted to all aspects of ﬁnite elements and wavelet methods in partial differential equations.

The topics include (but are not limited to): • adaptive methods, • parallel implementation, • high order methods. This year we particularly encourage talks on:

• high order FEM and isogeometrical analysis, • PDEs with fractional derivatives, • computational fluid mechanics.

Invited Speakers:

Mark Ainsworth (Brown University, Providence, USA) Volker John (WIAS and FU Berlin, Germany) Ricardo H. Nochetto (University of Maryland, College Park, USA) Gabriel Wittum (Goethe University Frankfurt, Germany, and KAUST, Saudi Arabia)

Conference Venue:

Bundesinstitut fur¨ Erwachsenenbildung (BifEB) Burglstein¨ 1-7 5360 St. Wolfgang / Strobl, Austria http://www.bifeb.at

Scientiﬁc Committee:

Th. Apel (Munchen),¨ S. Beuchler (Bonn), O. Ernst (Chemnitz), G. Haase (Graz), H. Harbrecht (Basel), R. Herzog (Chemnitz), M. Jung (Dresden), U. Langer (Linz), A. Meyer (Chemnitz), A. Rosch¨ (Duisburg-Essen), O. Steinbach (Graz)

Organising Committee:

U. Langer, W. Zulehner, M. Neumuller,¨ E. Lindner, A. Weihs, W. Forsthuber, S. Matculevich, S. Takacs, M. Weise, M. Pester, F. Ospald, A.-K. Glanzberg https://www.tu-chemnitz.de/mathematik/fem-symposium/ Programme 2 30th Chemnitz FEM Symposium 2017 – 10:50 ¨ urglsaal B Haupthaus Room: ...... 21 ...... 25 ...... 22 ...... 24 SR 2 : Svetozar Margenov Francisco-Javier Sayas Some new HDG Projections and theirfor Use Streamlined Analysis of HDG Methods. Oliver Rheinbach Domain Decomposition for Exascale Computing. Ariel Lombardi A Mixed Discretization of Elliptic Problems on Anisotropic Hybrid Meshes. Zoa de Wijn Mixed Finite Volume Element Methods for Three-field Formulations in Elasticity and Poroelasticity. Chair Room: - and h ...... 17 -Laplacian. p ...... 19 ...... 20 ¨ oder ...... 18 SR 1 : Jens Markus Melenk -Adaptive Mixed Finite Elements. Andreas Schr A Posteriori Error Estimates for hp Lothar Banz Higher Order FEM for the Obstacle Problem of the A Mixed Finite Element Approximation for the Compressible Euler Equations. Alex Bespalov Adaptive Stochastic Galerkin Methods for Parametric PDEs with Spatial Singularities. Herbert Egger Chair Room: ...... 14 ...... 12 ...... 13 ...... 16 ...... 15 ¨ oberl ¨ urglsaal B : Ulrich Langer : Thomas Apel Opening Gabriel Wittum Extreme Scale Solvers for Coupled Systems. Gundolf Haase FEM Completely Implemented for GPUs by Available Algorithm Libraries. Joachim Sch The Hellan-Herrmann-Johnson (HHJ) Method and the Tangential-Displacement Normal-Normal-Stress Continuous (TDNNS) Method. Sven Beuchler Michael Weise SPC-PM3AdH Finite Element Computations on Locally Refined Hexahedral Meshes with Hanging Nodes. A new Efficient Locking-free Mindlin–Reissner Plate Element. Tea and Coffee Break Morning Session Chair Session 65Chair Adaptive FEM I Advanced FEM Lunch Room: 9:30 9:40 11:15 12:30 11:40 12:05 10:30 10:50 Programme for Monday, September 25, 2017 30th Chemnitz FEM Symposium 2017 3 – 15:30 ¨ ¨ urglsaal urglsaal B B Room: Room: (continued) ...... 27 ...... 26 ...... 30 ...... 31 ...... 29 ...... 28 Tea and Coffee Break : Arnd Meyer : Stefan Takacs Adaptive Coupling of Finite Volume and Boundary Element Methods:Christoph Non-symmetric Hofer and Three-field FVM-BEM. Fast Multipatch Isogeometric Analysis Solvers. Daniel Jodlbauer Ricardo Nochetto Numerical Methods for Fractional Diffusion. Adaptive Mesh Refinement for Multiple Goal Functionals. Christoph Erath Parallel Block-Preconditioners for Fluid-Structure-Interaction Problems. Eglantina Kalluci The Parallel Implementation of the Hyperbolic Equation. Bernhard Endtmayer Afternoon Session Chair Fast Forward Postersession Chair followed by 14:50 14:00 Programme for Monday, September 25, 2017 4 30th Chemnitz FEM Symposium 2017 – 17:15 ...... 39 ...... 40 ...... 38 SR 2 : Walter Zulehner -smooth Isogeometric Functions 2 Mario Kapl C on Planar Multi-Patch Geometries. Svetlana Matculevich Adaptive IgA Based on the Functional-type Error Control. Ioannis Toulopoulos Time Discontinuous Galerkin Multipatch Isogeometric Analysis of Parabolic Problems. Chair Room: ...... 35 ...... 36 ...... 37 SR 1 : Sergej Rjasanow Vadim Korneev On the Accuracy and RobustnessPosteriori of Error A Majorants for Approximate Solutions of Reaction-Diffusion Equations. Maksim Frolov Reliability and Efficiency of Functional-type A Posteriori Error Estimates for Solid Mechanics in 2D:Comparison a of Standard and Mixed Finite Elements. Simon Becher On Layer-adapted Meshes for General Linear Turning Point Problems. Chair Room: (continued) ...... 34 ...... 32 ...... 33 ¨ urglsaal B : Ricardo Nochetto Svetozar Margenov Solution Methods for Fractional Diffusion Problems and Related Rational Approximations. Johannes Pfefferer hp-Finite Elements for Fractional Diffusion. Jens Markus Melenk Local FEMs for the Fractional Laplacian. Postersession / Tea and Coffee Break Fractional PDEs IChair Room: Adaptive FEM II Isogeometrical Analysis I 16:45 15:55 15:30 16:20 Programme for Monday, September 25, 2017 30th Chemnitz FEM Symposium 2017 5 SR 1 Haupthaus ...... 48 ...... 47 ...... 49 SR 2 : Michael Jung Roland Herzog Another GMRES please!?. Christoph Hofer Efficient Solvers for Discontinuous Galerkin Space Time Isogeometric Analysis Discretizations of Parabolic Problems. Stefan Takacs Robust Multigrid Methods for Isogeometric Discretizations of Multipatch Domains. Chair Room: ...... 44 ¨ auser ¨ oberl ...... 45 ...... 46 SR 1 : Joachim Sch Marius Paul Bruchh Goal-oriented Error Control for Stabilized Finite Element Methods. Fully Discrete A Posteriori Estimates for the Two-step Backward Differentiation Formula (BDF2) for the Time Dependent Stokes Equations. Adaptive Vertex-centered Finite Volume Methods (Petrov-Galerkin) with Convergence Rates for General Second-Order Linear Elliptic PDE. Andreas Brenner Christoph Erath Chair Room: (continued) ¨ oder ...... 41 ...... 43 ...... 42 ¨ urglsaal B : Andreas Schr Numerical Approximations of a Family of Nonlocal Operators on Bounded Domains. Nicole Cusimano Approximation of the Fractional Laplacian via hp-Finite Elements. Paolo Gatto Gabriel Acosta FEM for Fractional Evolution Problems. Fractional PDEs IIChair Adaptive FEM III Solvers Dinner / Get-together Meeting of the Scientific Committee Room: 17:15 18:30 17:40 18:05 20:00 Programme for Monday, September 25, 2017 6 30th Chemnitz FEM Symposium 2017 – 11:05 ¨ urglsaal B Room: ...... 56 ...... 55 ¨ uttler SR 2 : Bert J Felix Scholz Partial Tensor Decomposition for Decoupling Isogeometric Discretisations. Clemens Hofreither A Black-Box Algorithm for Fast Matrix Assembly in Isogeometric Analysis. Room: Chair ...... 53 ...... 54 SR 1 : Helmut Harbrecht Olaf Steinbach Coercive Space-time Finite Element Methods for Time-dependent Problems. Marco Zank Space-Time Methods for the Wave Equation. Room: Chair ...... 50 ...... 52 ...... 51 ¨ urglsaal B : Roland Herzog : Mark Ainsworth Fractional Cahn-Hilliard Equation(s): Analysis, Properties and Approximation. Chaobao Huang Mark Ainsworth Martin Stynes A New Analysis of a Numericalfor Method the Time-fractional Fokker-Planck Equation with General Forcing. Optimal Error Analysis of a Direct Discontinuous Galerkin Method for Time-fractional Reaction-Diffusion Equation. Room: Tea and Coffee Break Morning Session Chair Fractional PDEs IIIChair Space Time I Isogeometrical Analysis II 9:55 9:50 5 min break – switch to parallel session 9:00 10:45 10:20 Programme for Tuesday, September 26, 2017 30th Chemnitz FEM Symposium 2017 7 Haupthaus Haupthaus ...... 63 : ...... 65 ...... 64 options SR 2 : Martin Stynes Lutz Tobiska Local Projection Stabilization for a Convection-Diffusion Equation on a Surface. Axel Voigt A FEM Approach for aNavier-Stokes Surface Equation on Manifolds with Arbitrary Genus. Peter Gangl A Local Mesh Modification Strategy for Interface Problems with Application to Shape and Topology Optimization. Chair Room: Return ...... 60 ...... 61 ¨ uller ...... 62 SR 1 : Jens Lang Martin Neum A Time-parallel Algorithm for Parabolic Evolution Equations. Huidong Yang Monolithic Algebraic Multigrid Methods for a Space-time Finite Element Discretization of Parabolic Optimal Control Problems. Stefan Dohr Space-time Boundary Element Spaces and Operator Preconditioning for the Two-dimensional Heat Equation. Chair Room: (continued) (extra 45 Euro to pay cash on Monday) ...... 58 ...... 57 ...... 59 ¨ urglsaal B : Olaf Steinbach ¨ unther Of Numerical Solution of the General Diffusion Equation Based on the Boundary Element Methods and Chebyshev Approximation. Sergej Rjasanow G On the Non-symmetric FEM BEM Coupling for the Stokes Problem. Adaptive Wavelet Boundary Element Methods. Helmut Harbrecht BEMChair Room: Space Time IILunch Departure for Excursion FEM on Surfaces Conference Dinner 13:20 – Boat from Strobl14:50 to – St. Schafbergbahn Wolfgang from St. Wolfgang uphill to last stop Schafbergspitze (2) 17:05 railway / 17:40-19:00 walk to Strobl (1) 16:25 railway / 17:05 boat / 17:40 Strobl 11:55 11:05 12:20 13:05 19:30 11:30 Programme for Tuesday, September 26, 2017 8 30th Chemnitz FEM Symposium 2017 – 11:10 ¨ urglsaal B Room: ...... 72 ...... 71 SR 2 : Axel Voigt Jens Lang On the Stability and ConditioningAnisotropic of Finite-Element-Runge-Kutta Methods. Igor Voulis An Optimal Order DG Time Discretization Scheme for Parabolic Problems with Non-homogeneous Constraints. Chair Room: ...... 70 ...... 69 SR 1 : Vadim Korneev Thomas Apel Superconvergent Graded Meshes. Svetoslav Nakov Functional A Posteriori Error Estimates for the Nonlinear Poisson-Boltzmann Equation. Chair Room: ...... 68 ...... 67 ...... 66 ¨ urglsaal B : Lutz Tobiska : Volker John Volker John Finite Elements for Scalar Convection-Dominated Equations and Incompressible Flow Problems – a Never Ending Story. Alexander Linke Towards Pressure-robust Mixed Methods for the Incompressible Navier-Stokes Equations. Philip Lukas Lederer Pressure Robust Discretizations for Incompressible Flows. Morning Session Chair CFD IChair Adaptive FEMTea IV and Coffee Break Time Discretization Room: 9:55 9:50 5 min break – switch to parallel session 9:00 10:45 10:20 Programme for Wednesday, September 27, 2017 30th Chemnitz FEM Symposium 2017 9 ¨ urglsaal B Haupthaus Room: ...... 79 ...... 81 ...... 80 SR 2 : Gundolf Haase Matthias Hochsteger Automated Finite Element Assembling. Nicolas Neuss How to Make a CommonElement Lisp Library Finite High-performing?. Daniel Ganellari Domain Decomposition and Memory Footprint Reduction of an Eikonal Solver. Chair Room: ...... 76 ...... 78 ...... 77 SR 1 : Sven Beuchler (continued) Walter Zulehner A new Approach for Kirchhoff-Love Plates and Shells. Jan Petsche hp-FEM for a Stabilized Three-field Formulation of the Biharmonic Problem. Efficient Simulation of Short Fibre Reinforced Composites. Rolf Springer Chair Room: ...... 73 ...... 75 ...... 74 ¨ ollbacher ¨ urglsaal B : Gabriel Wittum ¨ urgen Fuhrmann Susanne H Relating FEM to FVM forProblems Interface in CFD. J A Coupled FEM-FVM Method for Electroosmotic Flow. Michael Neunteufel Fluid-Structure Interaction with H(div)-Conforming HDG and a new H(curl)-Conforming Method for Non-Linear Elasticity. Closing Lunch Departure CFD IIChair Solid Mechanics Efficient Implementation Room: 11:35 12:25 11:10 12:30 13:30 12:00 Programme for Wednesday, September 27, 2017

Collection of Abstracts 12 30th Chemnitz FEM Symposium 2017

Extreme Scale Solvers for Coupled Systems

Gabriel Wittum1 Arne Nagel¨ 2 Sebastian Reiter3

Numerical simulation has become one of the major topics in Computational Sci- ence. To promote modelling and simulation of complex problems new strategies are needed allowing for the solution of large, complex model systems. Crucial issues for such strategies are reliability, efficiency, robustness, usability, and versatility. After discussing the needs of large-scale simulation we point out basic simulation strategies such as adaptivity, parallelism and multigrid solvers. To allow adaptive, parallel computations the load balancing problem for dynamically changing grids has to be solved efficiently by fast heuristics. These strategies are combined in the simulation system UG (“Unstructured Grids”) being presented in the following. In the second part of the seminar we show the performance and efficiency of this strategy in various applications. In particular, large scale parallel computations of density-driven groundwater flow in het- erogenous porous media are discussed in more detail. Load balancing and efficiency of parallel adaptive computations is discussed and the benefit of combining parallelism and adaptivity is shown.

1 KAUST, ECRC, CEMSE, Thuwal, KSA, [email protected] 2 G-CSC, Universitat¨ Frankfurt, [email protected] 3 G-CSC, Universitat¨ Frankfurt, [email protected] 30th Chemnitz FEM Symposium 2017 13

FEM Completely Implemented for GPUs by Available Algorithm Libraries

Gundolf Haase1 Franz Pichler2

A ﬁnite element code is developed in which all computational expensive steps are performed on a graphics processing unit (GPU) via the THRUST and the PARALUTION library. The code is focused on simulation of transient problems where the repeated computations per time step create the computational cost. It is applied to solve partial and ordinary differential equations as they arise in thermal-runaway simulations of automotive batteries. The speedup obtained by utilizing the GPU for every critical step is compared against the single core and the multi-threading solution which is also supported by the chosen libraries. This way a high total speedup on the GPU is achieved without the need for programming a single classical Compute Uniﬁed Device Architec- ture (CUDA) kernel.

1 University of Graz, Institute for Mathematics and Scientiﬁc Computing, Graz, Austria, [email protected] 2 Virtual Vehicle Research Center, Graz, Austria, [email protected] 14 30th Chemnitz FEM Symposium 2017

The Hellan-Herrmann-Johnson (HHJ) Method and the Tangential-Displacement Normal-Normal-Stress Continuous (TDNNS) Method

Joachim Schoberl¨ 1 Astrid Pechstein2

The Hellan-Herrmann-Johnson (HHJ) method is a mixed ﬁnite element method to discretize Kirchhoff plate models. The tangential-displacement normal-normal-stress continuous (TDNNS) method is a method for the discretization of the elasticity equation. Both methods use matrix-valued normal-normal continuous ﬁnite element spaces for the momentum or stress variable, and also the bilinear-forms are tightly connnected. Based on the relation of these two methods, we propose new TDNNS spaces with less global degrees of freedom, and prove improved error estimates.

References:

[1] M. I. Comodi: The Hellan–Herrmann-Johnson Method: Some error estimates and postpro- cessing, Math. Comp. 52, 17–39, 1989 [2] A.S. Pechstein and J. Schoberl:¨ An analysis of the TDNNS method using natural norms, 2016, https://arxiv.org/abs/1606.06853. [3] D. Braess, A.S. Pechstein and J. Schoberl:¨ An Equilibration Based A Posteriori Error Estimate for the Biharmonic Equation and Two Finite Element Methods, 2017, https://arxiv.org/abs/1705.07607

1 TU Wien, Institute for Analysis and Scientiﬁc Computing, Vienna, Austria, [email protected] 2 Johannes Kepler University, [email protected] 30th Chemnitz FEM Symposium 2017 15

SPC-PM3AdH Finite Element Computations on Locally Reﬁned Hexahedral Meshes with Hanging Nodes

Sven Beuchler1

This contribution presents an overview about applications of the finite element pack- age SPC-PM3AdH, which was developed in the group of Arnd Meyer within the SFB 393 at the TU Chemnitz. The software has also been used and extended for high order finite element computations in three space dimensions. We show applications of the software • for adaptive finite element computations of scalar and vector valued problems, • with high order finite elements, • with boundary concentrated elements • to applications to optimal control.

1 INS, Uni Bonn, Wegelerstrasse 6, 53115 Bonn, [email protected] 16 30th Chemnitz FEM Symposium 2017

A new Efﬁcient Locking-free Mindlin–Reissner Plate Element

Michael Weise1

The Mindlin–Reissner plate model is widely used for the elastic deformation simulation of moderately thick plates. Shear locking occurs in the case of thin plates, which means slow convergence with respect to the mesh size. The Kirchhoff plate model does not show locking effects, but is valid only for thin plates. One would like to have a method suitable for both thick and thin plates. Several approaches are known to deal with the shear locking in the Mindlin–Reissner model. In addition to the well-known MITC elements and other approaches based on a mixed formulation, hierarchical methods have been developed in the recent years. A hierarchical deformation ansatz combining the Kirchhoff and Mindlin–Reissner models is given in [1]. An alternative rotation-free formulation for thick plates which is inherently locking-free was presented in [2]. A rotation-free isogeometric method for the original Mindlin–Reissner plate formulation was discussed in [3]. We present our new ﬁnite element formulation inspired by these references. The proposed element is not rotation-free, yet locking-free and performs very well in combination with a hierarchically preconditioned conjugate gradient method. Numerical comparisons between the mentioned elements are presented. A short outlook on an extension to Naghdi shells is also given.

References:

[1] R. Echter, B. Oesterle, M. Bischoff: A hierarchic family of isogeometric shell ﬁnite elements, Comput. Methods Appl. Mech. Engrg. 254 (2013) 170–180 [2] M. Endo, N. Kimura: An alternative formulation of the boundary value problem for the Timo- shenko beam and Mindlin plate, Journal of Sound and Vibration 301 (2007) 355–373 [3] B. Oesterle, E. Ramm, M. Bischoff: A shear deformable, rotation-free isogeometric shell formulation, Comp. Methods in Appl. Mech. and Engrg. 307 (2016) 235–255

1 TU Chemnitz, 09107 Chemnitz, Germany, [email protected] 30th Chemnitz FEM Symposium 2017 17

A Posteriori Error Estimates for h- and hp-Adaptive Mixed Finite Elements

Andreas Schroder¨ 1 Lothar Banz2 Jan Petsche3

In this talk, we present a posteriori error estimates and adaptivity of h- and hp- adaptive finite elements for mixed and mixed-hybrid methods, which are based on the introduction of flux or stress fields as additional unknowns in H(div)-spaces. In particular, we consider the Poisson problem and the obstacle problem, which lead to a variational equation and a variational inequality, respectively. The estimates rely on the use of post-processing reconstructions of the potential in H1 and, in the case of the obstacle problem, on the introduction of a certain Lagrange multiplier which is associated with the obstacle constraints. Two approaches of error control are discussed: In the first approach, the post-processing reconstruction is explicitly computed, whereas in the second approach, a reconstruction is applied which does not require an explicit computation. The latter enables the direct use of the discrete potential instead of its reconstruction, which significantly improves the error estimation. The applicability of the estimates is demonstrated in several numerical experiments, in which efficiency indices and convergence rates of h- and hp-adaptive schemes are studied.

References:

[1] J. Petsche, A. Schroder,¨ A posteriori error control and adaptivity of hp-ﬁnite elements for mixed and mixed-hybrid methods, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.05.032

1 University of Salzburg, Department of Mathematics, Salzburg, Austria, [email protected] 2 University of Salzburg, [email protected] 3 University of Salzburg, [email protected] 18 30th Chemnitz FEM Symposium 2017

Higher Order FEM for the Obstacle Problem of the p-Laplacian

Lothar Banz1 Bishnu P. Lamichhane2 Ernst P. Stephan3

We consider two higher order finite element discretizations of an obstacle problem with the p-Laplacian differential operator for p ∈ (1, ∞). The first approach is a nonlinear variational inequality in the primal variable u only. The second formulation is a primal-dual mixed formulation where the dual variable represents the signed residual of the variational inequality from the first approach. These two formulations are equivalent and, under mild assumptions on the obstacle, even on the discrete level when using biorthogonal basis functions. We prove a priori error estimates as well as a general a posteriori error estimate which is valid for both formulations. We present numerical results on the improved convergence rates of adaptive schemes (mesh size adaptivity with and without polynomial degree adaptation) for the singular case p = 1.5 and the degenerated case p = 3. We also present numerical results on the mesh indepen- dency and on the polynomial degree scaling of the discrete inf-sup constant when using biorthogonal basis functions for the dual variable defined on the same mesh with the same polynomial degree distribution.

1 University of Salzburg, Department of Mathematics, Salzburg, Austria, [email protected] 2 School of Mathematical & Physical Sciences, University of Newcastle, University Drive, Callaghan, NSW 2308, Aus- tralia, [email protected] 3 Institute of Applied Mathematics, Leibniz University Hannover, Welfengarten 1, 30167 Hannover, Germany, [email protected] 30th Chemnitz FEM Symposium 2017 19

Adaptive Stochastic Galerkin Methods for Parametric PDEs with Spatial Singularities

Alex Bespalov1 Leonardo Rocchi2

We consider a class of elliptic PDEs where the underlying differential operator has affine dependence on a large, possibly infinite, number of random parameters. Stochas- tic Galerkin Finite Element Methods (sGFEM) have emerged over the last decade as an efficient alternative to sampling methods for numerical solution of such problems. The sGFEM approximations are sought in tensor product spaces X ⊗ P , where X is a conventional finite element space associated with a physical domain and P is a finite- dimensional space of multivariate polynomials in the parameters. If a large number of random parameters is used to represent the input data and highly refined spatial grids are used for finite element approximations on the physical domain, then computing the sGFEM solution becomes prohibitively expensive, due to huge dimension of the space X ⊗ P . One way to avoid this is to use an adaptive approach, in which spatial (X-) and stochastic (P -) components of approximations are judiciously chosen and incrementally refined in the course of numerical computation. Adaptive refinement of spatial approximations is particularly important when solutions exhibit singularities (e.g., due to geometry of the physical domain). In this talk, we present an adaptive algorithm implementing stochastic Galerkin FEM. Building upon theoretical results in [1, 2] we employ a hierarchical a posteriori error estimation strategy to control the energy error and use the estimates of error reduction to steer adaptive refinement of spatial and stochastic components of Galerkin approximations. The results of numerical experiments for representative model problems with parametric coefficients and spatially singular solutions will be discussed. These results demonstrate the effectiveness of our adaptive refinement strategy.

References:

[1] A. Bespalov, C. E. Powell, and D. Silvester, Energy norm a posteriori error estimation for parametric operator equations, SIAM J. Sci. Comput., 36 (2014), pp. A339–A363. [2] A. Bespalov and D. Silvester, Efﬁcient adaptive stochastic Galerkin methods for parametric operator equations, SIAM J. Sci. Comput., 38 (2016), pp. A2118–A2140.

1 School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom, [email protected] 2 School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom, [email protected] 20 30th Chemnitz FEM Symposium 2017

A Mixed Finite Element Approximation for the Compressible Euler Equations

Herbert Egger1

We consider the flow of compressible fluids through pipes and pipe networks. As a starting point, a thermodynamically consistent variational characterization of solutions to the one dimensional Euler equations is presented which very directly encodes the conservation of mass, energy, and entropy. This variational principle is suitable for a conforming Galerkin approximation in space which automatically inherits the basic physical conservation laws. A mixed ﬁnite element method is briefly discussed as a particular choice. We also investigate the discretization in time by a problem adapted implicit time stepping scheme for which we prove exact conservation of mass and a slight dissipation of energy and negative entropy. These deviations from the strict conservation laws are due to numerical dissipation of the implicit time discretization. The resulting fully discrete method can be extended naturally to more general flow models and also to pipe networks and is therefore well suited for the simulation of gas transport in pipelines.

1 TU Darmstadt, Mathematics, Darmstadt, [email protected] 30th Chemnitz FEM Symposium 2017 21

Some new HDG Projections and their Use for Streamlined Analysis of HDG Methods

Francisco-Javier Sayas1 Shukai Du2 Allan Hungria3

The convergence analysis of the some of the Hybridizable Discontinuous Galerkin schemes was rendered ”quite trivial” by the introduction of a tailored HDG projection. This made many of the arguments in the analysis to be easy to replicate and verify. This special projection was not available in some new classes of HDG schemes, which required a certain amount of bootstrapping for the convergence proofs. We show that a carefully crafted (while easy to analyze) HDG projection allows for a very simple analysis of new HDG methods applied to diffusive and evolutionary problems.

1 University of Delaware, Mathematical Sciences, Newark DE, USA, [email protected] 2 University of Delaware, [email protected] 3 University of Delaware, [email protected] 22 30th Chemnitz FEM Symposium 2017

A Mixed Discretization of Elliptic Problems on Anisotropic Hybrid Meshes

Ariel Lombardi1 Alexis B. Jawtuschenko2

In this talk we consider the approximation in mixed form of elliptic problems in polyhedra. It is known that when the polyhedral domain is concave along edges or vertices, singularities may appear in the solution which degrade the numerical approximations. For the ﬁnite element method (FEM), strategies have been proposed in order to recover the optimal order of convergence, one of them being the use of meshes which are a priori adapted to the singularities. These meshes contain, in general, arbitrarily narrow elements, and as a consequence, the FEM tools to prove convergence have to manage this kind of elements. When mixed ﬁnite elements are considered, it is common the use of the H(div )-conforming Raviart-Thomas spaces on tetrahedral meshes to approximate the vectorial variable, and then, interpolation error estimates for the Raviart-Thomas interpolation operator are one of the main tools to analyse the approximation error. In order to prove optimal convergence of the mixed method, anisotropic interpolation error estimates like

3 ! X 2 2 2 ku − Π0ukL (T ) ≤ C(¯c) hik∂xi ukL (T ) + hT kdiv ukL (T ) , (1) i=1 are needed. Here, Π0 is the Raviart-Thomas interpolation operator of lowest order [Ned- elec, Raviart-Thomas], hT is the diameter of T and hi is the diameter of T in the xi- direction. This estimate was proved in [Acosta et al.] for a tetrahedron T with the constant C depending on the regular vertex property of T . When meshes contain arbitrarily anisotropic tetrahedra (this happens when the solution exhibits edge singularities) the constant C becomes arbitrarily large for some elements (known as slivers), as can be deduced from the results in [Acosta et al.]. As a consequence, optimal error estimates can not be obtained for this kind of approximation. For the simplest elliptic problem for the Laplace operator in polyhedra, with the aim to avoid the presence of slivers in anisotropic meshes, we propose a generalization of the standard mixed method mentioned before, which, in particular, allows for the use of hybrid meshes made up of triangularly right prisms, tetrahedra and pyramids. The meshes can contain arbitrarily anisotropic right prisms in order to deal with edge singularities, and isotropic tetrahedra to be able to consider general polyhedral domains. And (isotropic) pyramids are needed in order to glue right prisms and tetrahedra. For such a kind of meshes we introduce and analyse a mixed Finite/Virtual Element Method [Brezzi et al.]. The local discrete spaces coincide with the lowest order Raviart- Thomas spaces (and its extensions [Nedelec]) on tetrahedral and triangularly right pris- matic elements, and extend it to pyramidal elements. The local vectorial space on one

1 Facultad de Ciencias Exactas, Ingenier´ıa y Agrimensura, Universidad Nacional de Rosario, Departamento de Matematica,´ Rosario, Argentina, [email protected] 2 Departamento de Matematica,´ Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, [email protected] 30th Chemnitz FEM Symposium 2017 23 element E of any shape is deﬁned as n Vh(E) = v ∈ H(div ,E) ∩ H(curl ,E):

v · n ∈ P0(f) ∀f face of E, o div v ∈ P0(E), curl v = 0 where n denotes the exterior normal to E, and P0(S) the space of constant functions on S. As in the virtual element technology, the stiffness matrix can be computed from the known degrees of freedom of the shape functions. The discrete scheme is well posed and optimal error estimates are proved on meshes which allow for anisotropic elements. In particular, local interpolation error estimates for the virtual space are optimal and anisotropic on anisotropic right prisms, which can be used to obtain optimal approximation error estimates when the solution has edge or vertex singularities when suitably adapted meshes are used.

References:

[1] G. Acosta, Th. Apel, R.G. Duran,´ A.L. Lombardi. Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra. Math. Comp. 80#273 (2011) 141–163. [2] F. Brezzi, R.S. Falk, L.D. Marini. Basic principles of virtual element methods. Math. Model. Numer. Anal. 48 (2014), 1227–1240. [3] J.C. Ned´ el´ ec.´ Mixed ﬁnite elements in R3. Numer. Math. 35 (1980) 315–341. [4] P.A. Raviart, J.-M. Thomas. A mixed ﬁnite element method for second order elliptic problems, in Mathematical Aspects of the Finite Element Method, I. Galligani, E. Magenes, eds. Lectures Notes in Math. 606. Springer–Verlag 1977. 24 30th Chemnitz FEM Symposium 2017

Mixed Finite Volume Element Methods for Three-ﬁeld Formulations in Elasticity and Poroelasticity

Zoa de Wijn1 Ricardo Ruiz Baier2

We propose a family of mixed finite element and finite volume element methods for the approximation of linear elastostatics, formulated in terms of displacement, rotation vector, and solid pressure. We discuss the unique solvability of the continuous three- field formulation, as well as the invertibility and stability of the proposed Galerkin and Petrov-Galerkin formulations. Optimal a priori error estimates are derived using norms that are robust with respect to the Lame´ constants, turning these numerical methods to be particularly appealing for nearly incompressible materials. The predicted accuracy and applicability of the new three-field formulation and the corresponding mixed finite (volume) element schemes is verified numerically by conducting a number of computational tests in both 2D and 3D. Furthermore, we introduce a second order finite volume element formulation for a stationary three-field poroelasticity problem on a 2D domain, where we approximate the solid displacement, the pore-pressure of the fluid and an auxiliary unknown represent- ing the volumetric part of the total stress. The well-posedness of the proposed finite volume element method can only be established provided that suitable mesh geometric requirements are imposed on the family mesh partitions and it is also shown that the optimal error estimates are robust with respect to Lame’s´ first parameter approaching infinity, i.e. when the locking phenomenon occurs.

1 University of Oxford, Mathematical Institute, Oxford, United Kingdom, [email protected] 2 University of Oxford, Mathematical Institute, [email protected] 30th Chemnitz FEM Symposium 2017 25

Domain Decomposition for Exascale Computing

Oliver Rheinbach1

Preconditioned Newton-Krylov algorithms using scalable multilevel preconditioners from domain decomposition (DD) or multigrid (MG) have been the workhorse for the parallel solution of nonlinear implicit ﬁnite element problems for several decades. In these methods, the problem is ﬁrst linearized and then decomposed into parallel problems. On the contrary, in recent scalable nonlinear domain decomposition methods the nonlinear problem is directly decomposed into concurrent problems, i.e., before New- ton linearization. This approach increases the concurrency of the algorithm, reduces the need for synchronization and can also reduce energy consumption. We discuss recent nonlinear domain decomposition of the Nonlinear FETI-DP (Finite Element Tearing and Interconnecting) or Nonlinear BDDC (Balancing Domain Decompo- sition by Constraints) type to solve nonlinear hyperelasticity or plasticity problems and show parallel scalability to up to 800000 cores of the Mira BGQ supercomputer and 200000 cores of the Theta KNL supercomputer (both Argonne National Laboratory). We then apply these methods within a highly parallel two-scale numerical homogeniza- tion scheme using millions of MPI ranks for the simulation of micro-heterogeneous materials.

1 Technische Universitat¨ Bergakademie Freiberg, Mathematics and Computer Sciences , Germany, [email protected] 26 30th Chemnitz FEM Symposium 2017

Numerical Methods for Fractional Diffusion

Ricardo Nochetto1

We present three schemes for the numerical approximation of fractional diffusion, which build on different definitions of such a non-local process. The first method is a PDE approach that applies to the spectral definition and exploits the extension to one higher dimension. The second method is the integral formulation and deals with singular non-integrable kernels. The third method is a discretization of the Dunford-Taylor formula. We discuss pros and cons of each method, error estimates, and document their performance with a few numerical experiments.

1 University of Maryland, Institute for Physical Science and Technology [email protected] 30th Chemnitz FEM Symposium 2017 27

Adaptive Mesh Reﬁnement for Multiple Goal Functionals (Poster)

Bernhard Endtmayer1 Thomas Wick2 Ulrich Langer3

In this talk, we design a posteriori error estimation and mesh adaptivity for multiple goal functionals for elliptic problems. For this we use a dual-weighted residual approach in which localization is achieved in a variational form using a partition-of-unity. The key advantage is that the method is simple to implement and backward integration by parts is not required. For treating multiple goal functionals we employ the adjoint to the adjoint problem (i.e., a discrete error problem) and suggest an alternative way for its computation. Our algorithmic developments are substantiated for elliptic problems in terms of four different numerical tests that cover various types of challenges, such as singularities, different boundary conditions, and diverse goal functionals.

References:

[1] B. Endtmayer and T. Wick., A Partition-of-Unity Dual-Weighted Residual Approach for Multi- Objective Goal Functional Error Estimation Applied to Elliptic Problems. Computational Methods in Applied Mathematics, published online, doi:10.1515/cmam-2017-0001, 2017. [2] B. Endtmayer. Adaptive Mesh Reﬁnement for Multible Goal Functionals, Master thesis, Insti- tute of Computational Mathematics, JKU Linz, 2017.

1 Doctoral Program Computational Mathematics, Johannes Kepler University, Altenberger Straße 69, A-4040 Linz, Austria, [email protected] 2 Centre de Mathematiques´ Appliquees,´ Ecole´ Polytechnique , [email protected] 3 Institute of Computational Mathematics, [email protected] 28 30th Chemnitz FEM Symposium 2017

Adaptive Coupling of Finite Volume and Boundary Element Methods: Non-symmetric and Three-ﬁeld FVM-BEM (Poster)

Christoph Erath1 Robert Schorr2

We couple the prototype for flow and transport in porous media in an interior domain to the Laplace equation on the complement, an unbounded domain. This is a classical interface problem. An other interpretation of this model is that the (unbounded) exterior problem “replaces” the (unknown) boundary conditions of the interior problem. We approximate the solution of this interface problem either by the non-symmetric or the three-ﬁeld coupling of the Finite Volume Method (FVM) and the Boundary Element Method (BEM). For these two coupling methods, we introduce (semi-)robust a posteriori error estimators and use them in an adaptive algorithm to improve the convergence behavior. Numerical experiments compare these two adaptive methods in terms of effectivity indexes, errors and mesh reﬁnements. Both strategies turn out to be very suitable for the numerical treatment of interface problems, which have singularities or boundary/internal layers.

References:

[1] C. Erath. Coupling of the finite volume element method and the boundary element method: an a priori convergence result.SIAM J. Numer. Anal., 50(2): 574–594, 2012. [2] C. Erath and R. Schorr. An adaptive nonsymmetric finite volume and boundary element coupling method for a fluid mechanics interface problem.SIAM J. Sci. Comput., 39(3): A741–A760, 2017. [3] C. Erath. A posteriori error estimates and adaptive mesh refinement for the coupling of the finite volume method and the boundary element method.SIAM J. Numer. Anal. , 51(3): 1777– 1804, 2013. [4] C. Erath, G. Of, and F.-J. Sayas. A non-symmetric coupling of the finite volume method and the boundary element method.Numer. Math., 135: 895-922, 2017.

1 TU Darmstadt, Department of Mathematics, Dolivostr. 15, 64293 Darmstadt, Germany, [email protected] 2 TU Darmstadt, Department of Mathematics, Dolivostr. 15, 64293 Darmstadt, Germany, [email protected] 30th Chemnitz FEM Symposium 2017 29

Fast Multipatch Isogeometric Analysis Solvers (Poster)

Christoph Hofer1 Ulrich Langer2

In this contribution, we construct and investigate fast solvers for large-scale linear systems of algebraic equations arising from isogeometric analysis (IgA) of diffusion problems with heterogeneous diffusion coefficients on multipatch domains. In particular, we investigate the adaption of the Dual-Primal Finite Element Tearing and Intercon- necting (FETI-DP) method to IgA, called Dual-Primal IsogEometric Tearing and Inter- connecting (IETI-DP) method. We consider the cases of matching and non-matching meshes on the interfaces. In the latter case, we use a discontinuous Galerkin (dG) method to couple the different patches. This requires a special extension of the IETI-DP method to the dG-IgA formulation. We use ideas from the finite element case in order to formulate the corresponding IETI-DP method, called dG-IETI-DP. Furthermore, the method is extended to the case of non-matching interfaces due to incorrect segmentation, which produces gaps and overlaps in the domain decomposition. Numerical experiments show that the condition number κ behaves like O((1 + log(H/h))2), and is robust with respect to jumping diffusion coefficients and changing mesh-sizes across patch interfaces. We also study the dependence of κ on the underlying polynomial degree p of the NURBS used. In terms of p, we observe a logarithmic dependence. Moreover, we investigate the scaling behaviour of the classical IETI-DP method up to 1024 cores and present numerical results for complicated two and three dimensional domains. We investigate inexact versions utilizing multigrid methods for the solution of the patch- local problems. The advantage is a smaller memory footprint of the algorithm, hence, the possibility to solve larger systems. Finally, we present fast parallel solvers for the huge system arizing from stable space-time IgA approximations to parabolic diffusion problems. The solvers are based on time-parallel multigrid methods. Here the IETI-DP method is used as a part of the smoother.

1 Johannes Kepler University Linz, Doctoral Program ”Computational Mathematics”, Altenberger Straße 69, [email protected] 2 Johannes Kepler University Linz, Institute of Computational Mathematics, Altenberger Strasse 69, [email protected] 30 30th Chemnitz FEM Symposium 2017

Parallel Block-Preconditioners for Fluid-Structure-Interaction Problems (Poster)

Daniel Jodlbauer1 Ulrich Langer2 Thomas Wick3

The efﬁcient solution of nonlinear monolithic fluid-structure interaction problems is still a challenging problem. In this work, we present a preconditioner based on an approximate block LU-factorization for the solution of the arising linear systems. As shown in our previous work, this preconditioner shows robust behavior with respect to the mesh- and timestep-size and various material parameters. Additionally, we investigate the parallel performance of our solver and observe similar scalability results as [P. Crosetto, S. Deparis, G. Fourestey, A. Quarteroni. Parallel Algorithms for Fluid-Structure Interaction Problems in Haemodynamics, SIAM], being the only reference of monolithic scalability tests to our knowledge.

1 Johannes Kepler University, Doctoral Program Computational Mathematics, Altenbergerstrasse 69, 4040 Linz, [email protected] 2 Johannes Kepler University Linz, Institute for Computational Mathematics, [email protected] 3 Centre de Mathematiques´ Appliquees´ (CMAP) Ecole´ Polytechnique, [email protected] 30th Chemnitz FEM Symposium 2017 31

The Parallel Implementation of the Hyperbolic Equation (Poster)

Eglantina Kalluci1 Fatmir Hoxha2 Migert Xhaja3

In this paper we represent a parallel implementation, using MPI-programming of two schemes for solving the hyperbolic equation. Using an MPI programming we have implemented this problem in an asynchronous cluster with 9 processors, analysing the time of execution (= time of communication + time of computation) and the speedup. Another topic we have discussed is how these methods behave using OpenMP programming. The numerical tests are performed to ﬁnd the dimentions when the parallelization is more effective and at which platform. Key words: simultaneous methods, root, polynomial, parallel implementation, MPI, OpenMP.

References:

[1] Shuonan Dong. Finite Difference Methods for the Hyperbolic Wave, Partial Differential Equa- tions. [2] Eitan Tadmor. Spectral Methods for Hyperbolic Problems. [3] Scott Rostrup, Hans de Sterck. Parallel Hyperbolic PDE Simulation on Clusters. [4] Amy L. Shutz, Jim Lewis, Livia Miller. Hyperbolic Functions. [5] Danil Bykis, Patrick Mylan. Explicit Solutions of the Wave Equation on Three Dimensional Space-Times. [6] M. A. ARIGU, E.H. TWIZELL, A.B. GUMEL. Sequential and Parallel Methods for solving First- Order Hyperbolic Equations. [7] Qinghua Feng. Parallel Alternating Group Expilcit Iterative Method for Hyperboli Equations.

1 University of Tirana, Department of Applied Mathematics, Tirana, Albania, [email protected];[email protected] 2 University of Tirana, [email protected] 3 University Aleksander Moisiu, Durres, [email protected] 32 30th Chemnitz FEM Symposium 2017

Solution Methods for Fractional Diffusion Problems and Related Rational Approximations

Svetozar Margenov1

Our study is motivated by the recent achievements in fractional calculus and its nu- merous applications related to anomalous diffusion. Let us consider a fractional power of a self-adjoint elliptic operator introduced through its spectral decomposition, which is self-adjoint but nonlocal. The nonlocal problems are computationally expensive. Sev- eral different techniques were recently proposed to localize the nonlocal elliptic operator, thus increasing the space dimension of the original computational domain. An alternative approach is discussed in this talk. The goal is to reduce the computational complexity. Let A be a properly scaled symmetric and positive deﬁnite (SPD) sparse matrix. A method for solving algebraic systems of linear equations involving Aα, 0 < α ≤ 1 is presented. The solver is based on best uniform rational approximations (BURA) of the scalar functions tβ−α, 0 < t ≤ 1, β is a small integer. Although the fractional power of A is a dense matrix, the algorithm has complexity of order O(N), where N is the number of unknowns. Robust error estimates for the BURA based algorithm are obtained. A stable modiﬁcation of the Remez algorithm is developed to compute BURA for tβ−α. Two kinds of numerical experiments are presented. The 1D tests illustrate the sharpness of the error estimates, the positivity of the BURA based approximate inverse, as well as the mass conservation properties. The algorithm has optimal computational complexity, assuming that some optimal PCG solver is used to solve the involved auxiliary systems with certain positive diagonal perturbations of the original matrix A. The scalability analysis includes 3D tests with up to 5123 degrees of freedom. At the end, some promising parallel results on Intel Xeon Phi architecture towards scalability for extreme scale problems with fractional Laplacians are shown.

1 Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bontchev Str., Bl. 25A, 1113 Soﬁa, Bulgaria, [email protected] 30th Chemnitz FEM Symposium 2017 33

hp-Finite Elements for Fractional Diffusion

Johannes Pfefferer1 Dominik Meidner2 Boris Vexler3 Klemens Schurholz¨ 4

In this talk we introduce and analyze a numerical scheme based on hp-finite elements to solve boundary value problems involving the spectral fractional Laplacian. The approach is based on a reformulation of the problem posed on a semi-infinite cylinder in one more spatial dimension. After a suitable truncation of this cylinder, the resulting problem is discretized with linear finite elements in the original domain and with hp-finite elements in the extended direction. The proposed approach yields a reduction of the computational complexity in terms of degrees of freedom and even has slightly improved convergence properties compared to the state-of-the-art discretization using linear finite elements for both the original domain and the extended direction. The performance of the method is illustrated by numerical experiments.

1 Technical University of Munich, Garching, Germany, [email protected] 2 Technical University of Munich, [email protected] 3 Technical University of Munich, [email protected] 4 Technical University of Munich, [email protected] 34 30th Chemnitz FEM Symposium 2017

Local FEMs for the Fractional Laplacian

Jens Markus Melenk1 L. Banjai2 R. Nochetto3 E. Otarola4 A. Salgado5 C. Schwab6

We discuss several Finite Element Methods (FEMs) applied to the Caffarelli-Silvestre extension that localizes the fractional powers of symmetric, coercive,linear elliptic operators in bounded domains with Dirichlet boundary conditions. We consider open, bounded, polygonal not necessarily convex domains Ω ⊂ R2. First, we discretize with continuous, piecewise linear, Lagrangian FEM (P1-FEM) with mesh reﬁnement near corners, and prove that the full convergence rate can be attained. Second, we also prove that tensorization of a P1-FEM in Ω with a suitable hp-FEM in the extended variable achieves log-linear complexity with respect to the number of degrees of freedom in the domain Ω. Third, we propose a sparse tensor product FEM based on a multilevel P1-FEM in Ω and on a P1 FEM on radical–geometric meshes in the extended variable; this approach also achieves log-linear complexity with respect to NΩ. Fourth, under stronger (analyticity) assumptions on the data (including the geometry Ω), we establish exponential rates of convergence of hp-FEM for spectral, fractional diffusion operators by discretizing with high order elements.

1 TU Wien, Institute for Analysis and Scientiﬁc Computing, Wien, Austria, [email protected] 2 Heriot Watt University, [email protected] 3 University of Maryland, [email protected] 4 Universidad Federico Santa Maria, [email protected] 5 University of Tennessee, [email protected] 6 ETH Zurich, [email protected] 30th Chemnitz FEM Symposium 2017 35

On the Accuracy and Robustness of A Posteriori Error Majorants for Approximate Solutions of Reaction-Diffusion Equations

Vadim Korneev1

Dedicated to Ulrich Langer on his 65th birthday. Efficiency of the error control at numerical solutions of partial differential equations entirely depends on two factors: accuracy of an a posteriori error majorant and the computational cost of its evaluation, including the cost of a testing function or a vector field employed for the evaluation. In the paper, consistency of an a posteriori error bound implies that it is the same in the order with the respective unimprovable a priori bound and, therefore, it is a basic characteristic related to the first factor. The paper is dedicated to the elliptic reaction-diffusion equations, which are mod- elled by the equation −∆u + σu = f in Ω and u = 0 on the boundary ∂Ω. We present a guaranteed robust a posteriori error majorant which is effective at any constant reaction coefficient σ ≥ 0. Additionally, for the FEM (finite element method) solutions the majorant is consistent under assumptions that the mesh is quasiuniform and senior coefficients of the equation and the domain are sufficiently smooth. For big values of σ the majorant coincides with the majorant of Aubin (1972), the accuracy of which, as it is well known, deteriorates at σ tending to zero while at σ = 0 the majorant loses its 2 sense. For σ ∈ [0, σ∗] we upgrade Aubin’s majorant , where for FEM solutions σ∗ = ch and c = c(∂Ω) = const. In fact, we prove that for such σ the multiplier 1/σ, which in Aubin’s majorant stands before the square of L2-norm of the residual type term, can be replaced by ch2 with the same constant as above. Similar correction is applicable to a number of other known a posteriori error majorants with the residual type terms in the right parts. In particular it is applicable to the majorants for approximate solutions of the Poisson and reaction-diffusion equations which do not lose their sense at σ = 0, but nevertheless are not consistent with the a priori error bounds. The a posteriori error majorant developed in the paper can be expanded on a wider range of problems. In this relation we mention elliptic equations with piece wise constant nonnegative σ and elliptic equations of the 2n-th order, n ≥ 1. A part of the results is presented in the papers of Korneev (2016-2017) refered below. Research was supported by the grant from the Russian Fund of Basic Research, project N 15-01-08847a.`

References:

[1] V.G. Korneev. Consistent robust a posteriori error majorants for approximate solutions of diffusion-reaction equations. IOP Conf. Series: Materials Science and Engineering, 158, 2016, 012056. doi:10.1088/1757-899X/158/1/012056. [2] V.G. Korneev. Robust consistent a posteriori error majorants for approximate solutions of diffusion-reaction equations. arXiv:1702.00433v1 [math.NA] 1 Feb 2017. [3] V.G. Korneev. O tochnosti aposteriornyh funktsional’nyh mazhorant pogreshnosti priblizhen- nyh reshenii ellipticheskih uravnenii (On the accuracy of a posteriori functional error majorants for approximate solutions of elliptic equations), Doklady Academii Nauk. Matematika. (accepted for publication)

1 St. Petersburg State University, Department of mathematics and mechanics, St. Petersburg, Russia, [email protected] 36 30th Chemnitz FEM Symposium 2017

Reliability and Efﬁciency of Functional-type A Posteriori Error Estimates for Solid Mechanics in 2D: a Comparison of Standard and Mixed Finite Elements

Maksim Frolov1

This presentation is devoted to reliability and efficiency issues for functional approach (for example, see [1-3]) to a posteriori error control in 2D. We justify theoretically and confirm numerically that the approach yields reliable error bounds, which are valid for all conforming solutions of problems regardless of methods for solving. Error estimation requires construction of a set of new additional variables. It is shown that conforming finite element approximations in the Hilbert space H(div) for the additional variables provides a better choice for efficient implementations of the error control than standard finite elements. This work is supported by the Grant of the President of the Russian Federation MD-1071.2017.1.

References:

[1] S. Repin, A posteriori estimates for partial differential equations, Berlin, de Gruyter, 2008. [2] O. Mali, P. Neittaanmaki, S. Repin, Accuracy Veriﬁcation Methods. Theory and algorithms, Computational Methods in Applied Sciences, 32, Springer, 2014. [3] M. Churilova, M. Frolov, S. Repin, A posteriori error estimates for approximate solutions and adaptive algorithms for plane problems of elasticity theory, APM-2017 Proceedings (XLV In- ternational Conference ”Advanced Problems in Mechanics”, June 22-27, 2017, St. Petersburg, Russia), 2017.

1 Institute of Applied Mathematics and Mechanics, Department of Applied Mathematics, St. Petersburg, Russia, [email protected] 30th Chemnitz FEM Symposium 2017 37

On Layer-adapted Meshes for General Linear Turning Point Problems

Simon Becher1

We consider linear second order singularly perturbed boundary value problems with turning points on an interval. The number, location, and multiplicity of the turning points – zeros of the convection factor – is (almost) arbitrary. As result of the general setting, we have to be aware that different types of layers like exponential boundary layers, interior cusp-type layers, and certain power-type boundary layers may occur, see Liseikin (2001). In order to treat these layers and to enable uniform estimates, a convenient mesh construction strategy will be given which combines the well known Shishkin-type meshes with piecewise equidistant meshes proposed by Sun and Stynes (1994). In this talk we discuss the mesh construction and sketch how certain mesh properties can be used to prove uniform error estimates in the energy norm for higher order ﬁnite elements. The results are concretized for several examples with different layers. We also reveal that in general the energy norm is not balanced. This will be illustrated by some numerical experiments.

References:

[1] V. D. Liseikin, Layer resolving grids and transformations for singular perturbation problems, VSP, Utrecht, 2001. [2] G. Sun, M. Stynes, Finite element methods on piecewise equidistant meshes for interior turning point problems, Numer. Algorithms 8(1), 111–129, 1994. [3] S. Becher, Uniform error estimates for general semilinear turning point problems on layer- adapted meshes, arXiv:1701.06323v1 [math.NA], 2017.

1 Technische Universitat¨ Dresden, Institut fur¨ Numerische Mathematik, Dresden, Germany, [email protected] 38 30th Chemnitz FEM Symposium 2017

C2-smooth Isogeometric Functions on Planar Multi-Patch Geometries

Mario Kapl1 Vito Vitrih2

The space of C2-smooth isogeometric functions on bilinear planar multi-patch domains, where the graph of each isogeometric function is a multi-patch spline surface of bidegree (d, d), d = 5, 6, is considered. We investigate the dimension of the C2-smooth isogeometric space by decomposing the space into the direct sum of three simpler subspaces. Furthermore, we present an algorithm for the construction of a basis of the space, which is based on the concept of minimal determining sets for the involved spline coefﬁcients. Numerical results indicate that the resulting basis functions are well conditioned. The potential of the C2-smooth isogeometric space for applications in isogeometric analysis is demonstrated by solving the triharmonic equation, a sixth order partial differential equation, on different bilinear multi-patch domains. Moreover, we perform L2- approximation to experimentally show the optimal approximation power of the space. Finally, we describe possible extensions of the construction of C2-smooth isogeometric functions to more general domains.

1 Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Austria, Al- tenbergerstr. 69, 4040 Linz, Austria, [email protected] 2 IAM and FAMNIT, University of Primorska, Slovenia, [email protected] 30th Chemnitz FEM Symposium 2017 39

Adaptive IgA Based on the Functional-type Error Control

Svetlana Matculevich1 Ulrich Langer2 Sergey Repin3

We are concerned with adaptive IgA schemes applied to solve elliptic boundary value problems and initial boundary value problems of parabolic type. To provide guaranteed error control, we use functional error estimates (majorant and minorants) that are reliable, include only global constants (independent of the mesh characteristic h), and are valid for any approximation from the admissible functional space (see Repin, 1997 and Repin, 2002). For both static and evolutionary cases, we generate quantitatively efficient a posteriori error estimates and indicators by means of optimisation of the corresponding functionals. The efficient-in-time minimization (maximisation) of majorant (minorant) is achieved by performing it over higher smoothness approximation spaces defined on coarser grids. We confirm the reliability and efficiency of considered error estimates by performing uniform and adaptive refinement algorithms on a wide set of problems. The analysis of their performance is based on the results obtained from the extensive numerical testing.

References:

[1] S. Repin. A posteriori error estimation for nonlinear variational problems by duality theory. Zapiski Nauch. Sem. V. A. Steklov Math. Institute in St.-Petersburg (POMI), Vol. 243, 201-214, 1997. [2] S. I. Repin. Estimates of deviations from exact solutions of initial-boundary value problem for the heat equation. Rend., Mat. Acc. Lincei, 13(9):121-133, 2002.

1 Svetlana Matculevich, RICAM, Linz, Austria, [email protected] 2 Johann Radon Institute for Computational and Applied Mathematics (RICAM);, [email protected] 3 St. Petersburg Department of V.A. Steklov Institute of Mathematics RAS; University of Jyvaskyla, Finland, [email protected],[email protected] 40 30th Chemnitz FEM Symposium 2017

Time Discontinuous Galerkin Multipatch Isogeometric Analysis of Parabolic Problems

Ioannis Toulopoulos1 Christoph Hofer2 Ulrich Langer3 Martin Neumuller¨ 4

In this talk, we present a time discontinuous Galerkin multipatch Isogeometric Anal- ysis (dGIGA) scheme for solving linear parabolic problems. In our approach, we consider the time variable t as another variable, say xd+1, and the time derivative as a convection term in the direction xd+1. We derive the space-time weak formulation, by multiplying the Partial Differential Equation problem (PDE) by a test function depending on space and time variable. Using the resulting formulation, we deﬁne the dGIGA method. Pre- cisely, the whole space time cylinder is described as a union of space-time patches (slabs). In every space-time patch, the problem is simultaneously and uniformly discretized in space and in time, without imposing continuity requirements of the B-spline spaces across the interfaces of the space-time patches. The communication of the patch-wise discrete solutions is ensured by introducing simple “up-wind” jump terms across the interfaces. For stabilizing the time discretization, the method incorporates ideas of streamline diffusion methodology. We prove stability of the discrete problem with respect to a suitable norm, and show a priori discretization error estimates in this norm. The method has been implemented in a parallel platform. We present few numerical examples that support our theoretical estimates. This talk is based on the joint work [1]. This work was supported by the Austrian Science Fund (FWF) under the grant NFN S117-03 and W1214-N15, project DK4.

References:

[1] C. Hofer, U. Langer, M. Neumuller,¨ I. Toulopoulos, Time-Multipatch Discontinuous Galerkin Space-Time Isogeometric Analysis of Parabolic Evolution Problems (2017) under preparation.

1 Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Com- putational Methods for PDEs, Linz, Austria, [email protected] 2 Institute of Computational Mathematics, Johannes Kepler University (JKU) Linz, Austria, [email protected] 3 Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, and JKU Linz, Austria, [email protected] 4 Institute of Computational Mathematics, Johannes Kepler University (JKU) Linz, Austria, [email protected] 30th Chemnitz FEM Symposium 2017 41

Numerical Approximations of a Family of Nonlocal Operators on Bounded Domains

Nicole Cusimano1

The use of mathematical models involving fractional order derivatives to describe transport phenomena that deviate from the classical Markovian and Gaussian paradigm has been proposed in many different settings in the last few decades. A prototypical operator used to account for spatial nonlocality is the fractional Laplacian, which is naturally defined on unbounded domains. However, in many practical applications, the observed phenomena is confined to a particular bounded region in space. Hence, suitable modifications have to be made in the definition of the nonlocal operator in order to obtain a well-posed mathematical model, suitably accounting for the boundary conditions imposed by the physics of the observed problem. Motivated by the use of the spectral fractional Laplacian to describe spatial non locality on bounded domains, we study a discretisation method for this nonlocal operator based on a combination of the finite- element strategy and a quadrature formula for a singular one-dimensional integral. The proposed approach can be naturally applied to the case of more general nonlocal operators defined as fractional powers of their corresponding standard counterparts via their spectral expansion. A connection is made to fractional powers of non-negative definite matrices coming from the discretisation of the classical heat equation with suitable boundary conditions and some results for the considered approaches are shown.

1 Basque Center for Applied Mathematics, Mathematical Modelling in Biosciences, Bilbao, Spain, [email protected] 42 30th Chemnitz FEM Symposium 2017

Approximation of the Fractional Laplacian via hp-Finite Elements

Paolo Gatto1 Jan Hesthaven2

The fractional Laplacian operator on a bounded domain Ω can be realized as a Dirichlet-to-Neumann map for a degenerate elliptic equation posed in the semi-infinite cylinder Ω × (0, ∞). In fact, the Neumann trace on Ω involves a weight function that, according to the fractional exponent s, either vanishes (s < 1/2) or blows up (s > 1/2). On the other hand, the normal trace of the solution has the reverse behavior, thus making the Neumann trace analytically well-defined. Nevertheless, the solution develops an increasingly sharp boundary layer in the vicinity of Ω as s decreases. In this talk, I will discuss how to extend the technology of automatic hp-adaptivity, originally developed for standard elliptic equations, to accommodate for the problem of interest. I will conclude by showing an application of the fractional Laplacian to image denoising. In the image processing community, the standard way to apply the fractional Laplacian to a corrupted image is as a filter in Fourier space. This construction is inherently affected by the Gibbs phenomenon, which prevents the direct application to “spliced” images. Since our numerical approximation relies instead on the extension problem, it allows for processing different portions of a noisy image independently and combine them, without complications induced by the Gibbs phenomenon.

1 RWTH Aachen University, Schinkelstr. 2, D-52062 Aachen, Germany, [email protected] 2 Ecole Polytechnique Federale de Lausanne (EPFL), [email protected] 30th Chemnitz FEM Symposium 2017 43

FEM for Fractional Evolution Problems

Gabriel Acosta1 Francisco Bersteche2 Juan Pablo Borthagaray3

In this talk we introduce a ﬁnite element scheme for evolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time we consider is employed to represent memory effects, while a nonlocal differentiation operator in space accounts for long-range dispersion processes. We discuss well-posedness and obtain regularity estimates for the evolution problems under consideration. The discrete scheme we develop is based on piecewise linear elements for the space variable and a convolution quadrature for the time component. We present approximation error estimates for our method and illustrate its performance with numerical experiments in one- and two-dimensional domains.

1 University of Buenos Aires, Mathematics, Ciudad Autonoma´ de Buenos Aires, Argentina, [email protected] 2 University of Buenos Aires, [email protected] 3 University of Buenos Aires, [email protected] 44 30th Chemnitz FEM Symposium 2017

Goal-oriented Error Control for Stabilized Finite Element Methods

Marius Paul Bruchhauser¨ 1 Markus Bause2 Kristina Schwegler3

The numerical approximation of nonstationary convection-diffusion-reaction problems ∂tu + b · ∇u − ∇ · (ε∇u) + αu = f (1) with small diffusion 0 < ε 1 remains to be a challenging task. Eq. (1) is considered as a prototype model for more sophisticated equations of practical interest. For its numerical solution stabilized methods like the SUPG approach are used that aim to introduce a correct amount of artificial diffusion in regions with sharp inner or boundary layers or complicated structures where important physical or chemical phenomena take place. Even though it seems to be natural to combine stabilized finite element methods with adaptive error control mechanisms to further enhance the approximation quality, this combination has been studied rarely so far in the literature. Existing a posteriori error analyses are either typically based on error norms that are non natural for the stabilized scheme or they are not robust with respect to the small diffusion parameter, i.e., that they involve constants that increase for vanishing diffusion. In this contribution we combine stabilized finite element methods with an a posteriori error control mechanism based on a dual weighted residual approach. The dual weighted error estimator assesses the discretization error with respect to a given goal quantity of physical interest. In contrast to former works on goal-oriented error control for transport problems, we solve the stabilized dual problem by a higher order approach and do not use a computationally less expensive higher order interpolation technique to determine the approximate dual solution. This is done in order to improve the approximation quality of the dual solution in the sensitive regions, i.e., in layers and regions with steep gradients. Thereby we aim to get an error representation for the goal quantity to the best feasible extent rather than an a posteriori error estimation. The derivation of our goal-oriented error control for SUPG stabilized approximations of Eq. (1) is presented. Moreover, its numerical performance properties are studied and illustrated for benchmark problems of convection-dominated transport.

References:

[1] M. Bause, M. Bruchhauser,¨ K. Schwegler, A goal-oriented a posteriori error control for unsteady convection-dominated problems, to appear, 2017.

1 Helmut Schmidt University, Faculty of Mechanical Engineering, Hamburg, Germany, [email protected] 2 Helmut Schmidt University, Faculty of Mechanical Engineering, Hamburg, Germany, [email protected] 3 Helmut Schmidt University, Faculty of Mechanical Engineering, Hamburg, Germany, [email protected] 30th Chemnitz FEM Symposium 2017 45

Fully Discrete A Posteriori Estimates for the Two-step Backward Differentiation Formula (BDF2) for the Time Dependent Stokes Equations

Andreas Brenner1 Eberhard Bansch¨ 2

We present optimal order residual-based a posteriori error estimates for the fully discrete instationary Stokes equations. The time discretization uses the two-step backward differential formula method (BDF2) and the space discretization is based on inf- sup stable pairs of finite elements, where we allow arbitrary mesh changes and variable time steps. An algorithm for variable time steps is presented and computational examples are given to confirm the theoretical findings.

1 University Erlangen, Applied Mathematics III, Cauerstr. 11, 91058 Erlangen, Germany, [email protected] 2 University Erlangen, Applied Mathematics III, [email protected] 46 30th Chemnitz FEM Symposium 2017

Adaptive Vertex-centered Finite Volume Methods (Petrov-Galerkin) with Convergence Rates for General Second-Order Linear Elliptic PDE

Christoph Erath1 Dirk Praetorius2

A classical finite volume method describes numerically a conservation law of an underlying model problem. It naturally preserves local conservation of the numerical fluxes. Therefore, finite volume methods are well-established in the engineering community (fluid mechanics). In this lecture, we approximate the solution of a general second-order linear elliptic PDE with an adaptive vertex-centered finite volume method (FVM). Note that we can write the vertex-centered FVM with first-order conforming ansatz functions on a primal mesh and piecewise constant test functions on the corresponding dual mesh also in variational form (Petrov-Galerkin method). The adaptive mesh-refinement is driven by the local contributions of a weighted-residual error estimator. We prove that the adaptive algorithm leads to linear convergence with generically optimal algebraic rates for the error estimator and the sum of energy error plus data oscillations. Similar results have been derived for finite element methods and boundary element methods. However, the lack of the classical Galerkin orthogonality for FVM leads to new challenges. For non-symmetric model problem configurations, we additionally have to prove a new L2-type estimate.

References:

[1] C. Erath and D. Praetorius. Adaptive vertex-centered ﬁnite volume methods with convergence rates. SIAM J. Numer. Anal., 54(4):2228–2255, 2016. [2] C. Erath and D. Praetorius. Cea-type´ quasi-optimality and convergence rates for (adaptive) vertex-centered FVM.Finite Volumes for Complex Applications VIII - Methods and Theoretical As- pects, Springer Proceedings in Mathematics & Statistics, 199:215–223, 2017. [3] C. Erath and D. Praetorius. Convergence rates of adaptive vertex-centered ﬁnite volume methods for general second order linear elliptic PDEs.Preprint, August 2017.

1 TU Darmstadt, Department of Mathematics, Darmstadt, Germany, [email protected] 2 TU Wien, Institute for Analysis and Scientiﬁc Computing, Wiedner Hauptst. 8-10, 1040 Wien, Austria, [email protected] 30th Chemnitz FEM Symposium 2017 47

Another GMRES please!?

Roland Herzog1

In October 2013 Arnd Meyer gave a talk in the numerics research seminar at TU Chemnitz discussing general Krylov subspace methods which would employ a preconditioner and two individual inner products. One inner product would be used in the Arnoldi process for the generation of the Krylov subspace’s orthonormal basis. The second inner product deﬁnes the norm in which the error is minimized in each iteration. Arnd gave an overview over potential combinations of preconditioners and inner products and suggested new variants of known methods, in particular of the gener- alized minimal residual method (GMRES). In this presentation we follow up on these ideas and discuss a general version of GMRES from a Hilbert space perspective. The talk will reveal whether or not versions of GMRES have been overlooked and ’another GMRES’ is needed.

1 TU Chemnitz, Mathematics, Chemnitz, Germany, [email protected] 48 30th Chemnitz FEM Symposium 2017

Efﬁcient Solvers for Discontinuous Galerkin Space Time Isogeometric Analysis Discretizations of Parabolic Problems

Christoph Hofer1 Ulrich Langer2

In this talk, we construct and investigate fast solvers for large-scale linear systems of algebraic equations arising from the application of isogeometric analysis (IgA) to parabolic diffusion problems. We consider decompositions of the space time cylinder into time slabs, where each slab is again decomposed into several space-time patches. We use dG-techniques to provide information transfer between the time slabs, whereas the patches in a time slab are coupled in a conforming way. In the ﬁrst part of the talk, we present the assembling techniques, which is based on the tensor product structure of the problem. The second part deals with the solution strategy, which is based on the time parallel multigrid method developed in [1]. We utilize the multipatch structure of the time slabs by using some iterations of the Isogeometric Tearing and Interconnecting method as smoother in each time slab. We conclude the talk the discussion of some numerical results.

References:

[1] M. J. Gander and M. Neumueller. Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems, SIAM Journal on Scientiﬁc Computing 2016 38:4, A2173-A2208

1 Johannes Kepler University Linz, Doctoral Program Computational Mathematics, Austria, [email protected] 2 Johannes Kepler University Linz, Institute for Computational Mathematics, Austria, [email protected] 30th Chemnitz FEM Symposium 2017 49

Robust Multigrid Methods for Isogeometric Discretizations of Multipatch Domains

Stefan Takacs1

Isogeometric Analysis (IgA) is a novel approach for the discretization of partial differential equations, which is based on B-spline or NURBS ansatz functions and a representation of the computational domain by a global geometry function. More complicated domains cannot be represented by just one such geometry function. Instead, the whole domain is decomposed into patches, where each of those is represented by its own geometry function. In IgA, we typically encounter as discretization parameters the mesh size and the spline degree. If linear solvers from standard ﬁnite elements are transferred to IgA in a naive way, typically their behaviour in the mesh size is as good as in the ﬁnite element case, but the performance deteriorates if the spline degree is increased. The same holds for multigrid solvers, where robustness in the grid size is not an issue, but standard smoothers (like Gauss Seidel) suffer from the exponential growth of the condition number of the linear system in the spline degree. For the single patch case, the author and his coworkers have proposed a multigrid solver being robust both the grid size and the spline space. The solver exploits the tensor-product structure of the problem and robust approximation error and inverse estimates. In the talk we will see how domain decomposition approaches can be used to extend that smoother to the multipatch case. This yields methods showing robust convergence behaviour in the grid size and the spline degree. We will discuss how to develop a convergence theory that yields explicit bounds in terms of the grid size and the spline degree.

1 Austrian Academy of Sciences, RICAM, Linz, Austria, [email protected] 50 30th Chemnitz FEM Symposium 2017

Fractional Cahn-Hilliard Equation(s): Analysis, Properties and Approximation

Mark Ainsworth1 Zhiping Mao2

The classical Cahn-Hilliard equation [1] is a non-linear, fourth order in space, parabolic partial differential equation which is often used as a diffuse interface model for the phase separation of a binary alloy. Despite the widespread adoption of the model, there are good reasons for preferring models in which fractional spatial derivatives appear [2,3]. We consider two such Fractional Cahn-Hilliard equations (FCHE). The ﬁrst [4] corresponds to considering a gradient flow of the free energy functional in a negative order Sobolev space H−α , α ∈ [0, 1] where the choice α = 1 corresponds to the classical Cahn-Hilliard equation whilst the choice α = 0 recovers the Allen-Cahn equation. It is shown that the equation preserves mass for all positive values of fractional order and that it indeed reduces the free energy. The well-posedness of the problem is established in the sense that the H1-norm of the solution remains uniformly bounded. We then turn to the delicate question of the L∞ boundedness of the solution and establish an L∞ bound for the FCHE in the case where the non-linearity is a quartic polynomial. As a consequence of the estimates, we are able to show that the Fourier-Galerkin method delivers a spectral rate of convergence for the FCHE in the case of a semi-discrete approximation scheme. Finally, we present results obtained using computational simulation of the FCHE for a variety of choices of fractional order α. We then consider an alternative FCHE [3,5] in which the free energy functional involves a fractional order derivative. References:

[1] J.W. Cahn and J.E. Hilliard, Free energy of a non-uniform system. I. Interfacial Free Energy, J. Chem. Phys, 28, 258–267 (1958) [2] L. Caffarelli and E. Valdinoci, A Priori Bounds for solutions of non-local evoluation PDE, Springer, Milan 2013. [3] G. Palatucci and O. Savin, Local and global minimisers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl., 4, 673–718 (2014). [4] M. Ainsworth and Z. Mao, Analysis and Approximation of a Fractional Cahn-Hilliard Equation, (In review, 2016). [5] M. Ainsworth and Z. Mao, Well-posedness of the Cahn-Hilliard Equation with Fractional Free Energy and Its Fourier-Galerkin Discretization, (In review, 2017).

1 Division of Applied Mathematics, Brown University, Providence RI 02912, USA, [email protected] 2 Division of Applied Mathematics, Brown University, Providence RI 02912, USA, [email protected] 30th Chemnitz FEM Symposium 2017 51

A New Analysis of a Numerical Method for the Time-fractional Fokker-Planck Equation with General Forcing

Martin Stynes1 Can Huang2 Kim Ngan Le3

Two new convergence analyses are given for the ﬁnite element spatial discretization and piecewise-constant time discretization scheme that is used in [K. Le et al., SIAM J. Numer. Anal., (54) 2016, pp.1763–1784] to solve the time-fractional Fokker-Planck equation on a domain Ω × [0,T ] with general forcing, i.e., where the forcing term is a function of both space and time. First, when the method is discretised only in space, stability and convergence are proved in a fractional norm that is stronger than the L2(Ω) norm used in the above paper. Furthermore, unlike the bounds proved in Le et al., the constant multipliers in our analysis do not blow up as the order of the fractional derivative α approaches the classical value of 1. Second, when the method is discretised only in time, we present a new L2(Ω) convergence proof that avoids a flaw in the proof of Theorem 4.4 of the Le et al. paper.

1 Beijing Computational Science Research Center, Beijing 100193, China, [email protected] 2 School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Perfor- mance Scientiﬁc Computing, Xiamen University, Fujian 361005, China, [email protected] 3 School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia, [email protected] 52 30th Chemnitz FEM Symposium 2017

Optimal Error Analysis of a Direct Discontinuous Galerkin Method for Time-fractional Reaction-Diffusion Equation

Chaobao Huang1 Martin Stynes Na An

A time-dependent reaction-diffusion initial-boundary value problem with a Caputo time derivative of order α ∈ (0, 1) is considered. Its solution has a weak singularity at the initial time t = 0. Bounds on certain derivatives of the solution are obtained. A fully discrete Direct Discontinuous Galerkin (DDG) method that is designed to deal with this initial singularity is presented and analysed. In this method the well-known L1 scheme on a graded mesh is used for the time discretisation, while a DDG method on a uniform mesh is used in the spatial direction. Then L2-norm stability and consistency estimates are derived for the method; during this analysis, a new projection is developed to handle the Dirichlet boundary conditions—it is shown to be well deﬁned (which is a non-trivial result) and bounds on the projection error are derived in various norms. Using this information, an optimal L2-norm error estimate is obtained. Numerical experiments are presented that conﬁrm the sharpness of the error analysis.

1 Beijing Computational Science Research Center, Applied and Computational Mathematics Division, Beijing, China, [email protected] 30th Chemnitz FEM Symposium 2017 53

Coercive Space-time Finite Element Methods for Time-dependent Problems

Olaf Steinbach1

For time-dependent partial differential equations such as the heat or wave equation we discuss variational formulations which turn out to satisfy a related stability condition, or an equivalent ellipticity estimate. We provide a stability and error analysis and we present some numerical results which conﬁrm the theoretical ﬁndings. The talk is based on joint work with M. Zank.

1 TU Graz, Institut fur¨ Numerische Mathematik, Graz, Austria, [email protected] 54 30th Chemnitz FEM Symposium 2017

Space-time Methods for the Wave Equation

Marco Zank1 Olaf Steinbach2

For the discretisation of time-dependent partial differential equations usually explicit or implicit time stepping schemes are used. An alternative approach is the usage of space-time methods, where the space-time domain is discretised and the resulting global linear system is solved at once. In this talk the model problem is the scalar wave equation. First, a brief overview of known results for the wave equation is presented. Second, a space-time formulation is motivated and discussed. Finally, numerical examples for a one-dimensional spatial domain are presented.

1 TU Graz, Institut fur¨ Numerische Mathematik, Graz, Austria, [email protected] 2 TU Graz, [email protected] 30th Chemnitz FEM Symposium 2017 55

Partial Tensor Decomposition for Decoupling Isogeometric Discretisations

Felix Scholz1 Angelos Mantzaflaris2 Bert Juttler¨ 3

In isogeometric analysis, tensor decomposition methods can be applied to overcome the computational difficulties when performing the quadrature for assembling the system matrix. Unlike the discretisations used in finite element methods, the spline discretisations employed in isogeometric analysis possess a global tensor product structure which can be used in several ways to reduce the complexity of the quadrature. The exploitation of this tensor product structure enables us to deal with the computational disadvantages stemming from the increased polynomial degrees and the larger support of the basis functions In the present work we introduce a partial tensor decomposition based on singular value decomposition which is applied to the integrands’ coefficient tensors in an isogeometric discretisation, thereby replacing the trivariate or fourvariate quadrature by the evaluation of a number of lower-variate integrals. In three dimensions, the method achieves quasi-optimal computational complexity for the assembly of the system matrices and outperforms the assembly method obtained by using a full tensor decomposition of the coefficient tensors. A natural application of this approach are four- dimensional space-time problems, making it possible to decouple the integration in space from the integration in time. We analyse the computational complexity of the method and demonstrate its ad- vantageous behaviour both theoretically and in the run-times of computationally de- manding numerical experiments.

1 Johann Radon Institute for Computational and Applied Mathematics, Altenberger Straße 69, 4040 Linz, Austria, [email protected] 2 Johann Radon Institute for Computational and Applied Mathematics, [email protected] 3 Johann Radon Institute for Computational and Applied Mathematics, Institute of Applied Geometry, Johannes Ke- pler Universitat¨ Linz, [email protected] 56 30th Chemnitz FEM Symposium 2017

A Black-Box Algorithm for Fast Matrix Assembly in Isogeometric Analysis

Clemens Hofreither1

A fast algorithm for assembling stiffness matrices for Isogeometric Analysis with tensor product spline spaces is presented. The procedure exploits the facts that (a) such matrices have block-banded structure, and (b) they often have low Kronecker rank. Combined, these two properties allow us to reorder the nonzero entries of the stiffness matrix into a relatively small, dense matrix or tensor of low rank. A suitable black-box low-rank approximation algorithm is then applied to this matrix or tensor. This allows us to approximate the nonzero entries of the stiffness matrix while explicitly computing only relatively few of them, leading to a fast assembly procedure. The algorithm does not require any further knowledge of the used spline spaces, the geometry transform, or the partial differential equation, and thus is black-box in nature. Existing assembling routines can be reused with minor modiﬁcations. A reference implementation is provided which can be integrated into existing code. In several numerical examples, we demonstrate signiﬁcant speedups over a standard Gauss quadrature assembler for several geometries in two and three dimensions.

1 Johannes Kepler University Linz, Institute of Computational Mathematics, Linz, Austria, [email protected] 30th Chemnitz FEM Symposium 2017 57

Numerical Solution of the General Diffusion Equation Based on the Boundary Element Methods and Chebyshev Approximation

Sergej Rjasanow1 Steffen Weißer2

A combination of the Boundary Element Methods (BEM) and Chebyshev approximation is applied to the three dimensional Dirichlet boundary value problem for the general diffusion equation with variable matrix-valued coefﬁcient. The advantages of both methods lead to an iterative procedure which converges independently of the discretisation parameters. The volume mesh is avoided, only a BEM surface discretisation and the tensor product Chebyshev mesh are used. Some numerical examples illustrate the efﬁciency of this combination of two numerical methods on hand of some analytically known solutions as well as FEM reference solutions.

1 Saarland University, Mathematics, Saarbrucken,¨ Germany, [email protected] 2 Saarland University, Mathematics, Saarbrucken,¨ Germany, [email protected] 58 30th Chemnitz FEM Symposium 2017

Adaptive Wavelet Boundary Element Methods

Helmut Harbrecht1 Manuela Utzinger2

This talk is concerned with numerical techniques for the adaptive application of global operators of potential type in wavelet coordinates. This is a core ingredient for a new type of adaptive solvers that has so far been explored primarily for PDEs. We shall show how to realize asymptotically optimal complexity in the present context of global operators. Asymptotically optimal means here that any target accuracy can be achieved at a computational expense that stays proportional to the number of degrees of freedom (within the setting determined by an underlying wavelet basis) that would ideally be necessary for realizing that target accuracy if full knowledge about the unknown solution were given. The theoretical findings are supported and quantified by first numerical experiments.

References:

[1] W. Dahmen, H. Harbrecht, and R. Schneider. Adaptive methods for boundary integral equations. Complexity and convergence estimates. Math. Comput., 76(259):1243-1274, 2007. [2] H. Harbrecht and M. Utzinger. On adaptive wavelet boundary element methods. Preprint 2015-42, Mathematisches Institut, Universitat¨ Basel, Switzerland, 2015 (to appear in J. Comput. Math.).

1 University of Basel, Department of Mathematics and Computer Science, Basel, Switzerland, [email protected] 2 University of Basel, Department of Mathematics and Computer Science, [email protected] 30th Chemnitz FEM Symposium 2017 59

On the Non-symmetric FEM BEM Coupling for the Stokes Problem

Gunther¨ Of1

In recent years, there has been substantial progress on the stability analysis of discrete systems of the so-called non-symmetric coupling of FEM and BEM for Lipschitz domains. In this setting a ﬁnite element discretization is used for one subdomain while the weakly singular boundary integral equation is considered for a second subdomain. This approach leads to non-symmetric discrete linear systems. In this talk, we will discuss the non-symmetric coupling for the Stokes problem, which includes additional challenges, and we will present related numerical examples.

1 Graz University of Technology, Institute of Computational Mathematics, Graz, Austria, [email protected] 60 30th Chemnitz FEM Symposium 2017

A Time-parallel Algorithm for Parabolic Evolution Equations

Martin Neumuller¨ 1 Iain Smears2

We present an original time-parallel algorithm for the solution of the implicit Euler discretization of general parabolic evolution equations with self-adjoint spatial operators. The main features of the proposed algorithm include a strong decoupling between time and space, a detailed convergence theory for time-dependent spatial operators, and a parallel complexity per iteration that depends only logarithmically on the total number of time-steps. Furthermore ﬁrst numerical experiments will be presented.

1 Johannes Kepler University Linz, Institute of Computational Mathematics, Linz, Austria, [email protected] 2 INRIA Paris, 2 Rue Simone Iff, 75589 Paris, France, [email protected] 30th Chemnitz FEM Symposium 2017 61

Monolithic Algebraic Multigrid Methods for a Space–time Finite Element Discretization of Parabolic Optimal Control Problems

Huidong Yang1 Olaf Steinbach 2

In this talk, we will present some numerical studies on monolithic algebraic multigrid methods for solving the linear system of algebraic equations arising from a space–time finite element discretization of parabolic optimal control problems. The finite element discretization is based on the recent results [O. Steinbach: Space–time finite element methods for parabolic problems, Comput. Methods Appl. Math., 15:551–566, 2015]. We will mainly focus on robustness of the monolithic algebraic multigrid methods for solving the discretized optimal control problems all at once, that are based on the algebraic multigrid methods we have recently developed for the space–time finite element discretization of parabolic problems.

1 Institut fur¨ Numerische Mathematik, Technische Universitat¨ Graz, Graz, Austria, [email protected] 2 Institut fur¨ Numerische Mathematik, Technische Universitat¨ Graz , [email protected] 62 30th Chemnitz FEM Symposium 2017

Space-time Boundary Element Spaces and Operator Preconditioning for the Two-dimensional Heat Equation

Stefan Dohr1 Olaf Steinbach2

The standard approach in space-time boundary element methods for discretizing variational formulations of boundary integral equations is using space-time tensor product spaces originating from a separate decomposition of the boundary Γ and the time interval (0,T ). However, this approach does not allow adaptive refinement in space and time simultaneously. This motivates the use of an arbitrary decomposition of the whole space-time boundary Σ = Γ × (0,T ) into boundary elements. In this talk we consider the two-dimensional heat equation as a model problem and compare these two discretization methods. Moreover, when using space-time tensor product spaces we can construct a preconditioner for the first boundary integral equation by using the discretization of the hypersingular operator with respect to an appropriate dual mesh. The theoretical results are confirmed by numerical tests.

1 TU Graz, Institut fur¨ Numerische Mathematik, Graz, Austria, [email protected] 2 TU Graz, [email protected] 30th Chemnitz FEM Symposium 2017 63

Local Projection Stabilization for a Convection-Diffusion Equation on a Surface

Lutz Tobiska1 Kristin Simon2

We consider the convection-diffusion equation

−ε∆Γu + b · ∇Γu + cu = f on Γ posed on a hypersurface Γ. Here, ∇Γ and ∆Γ denote the surface gradient and the Laplace-Beltrami operator, respectively. The assumption 1 c(x) − (∇ · b)(x) ≥ σ > 0 for all x ∈ Γ 2 Γ 0 guarantees the unique solvability of the associated weak formulation of the problem. As known for this type of equations when posed in a domain Ω ⊂ Rd, d = 2, 3, with boundary conditions, boundary and interior layers may occur and standard finite element methods tend to be unstable unless the mesh is sufficiently fine. Many approaches have been developed and studied to overcome these instabilities. In case of a transport equation on a closed surface Γ much less is known. We propose a one-level local projection type stabilization and give an a priori error analysis in a mesh-dependent norm with error constants independent of ε.

1 Institute for Analysis and Computational Mathematics, Otto von Guericke Magdeburg, Magdeburg, Germany, [email protected] 2 Institute for Analysis and Computational Mathematics, Otto von Guericke Magdeburg, Magdeburg, Germany, [email protected] 64 30th Chemnitz FEM Symposium 2017

A FEM Approach for a Surface Navier-Stokes Equation on Manifolds with Arbitrary Genus

Axel Voigt1 Sebastian Reuther2

We consider a compact smooth Riemannian surface S without boundary and an incompressible surface Navier-Stokes equation 1 ∂ v + ∇ v = −∇ p + −∆dRv + 2κv t v S Re S ∇S · p = 0 in S × (0, ∞) with initial condition v (x, t = 0) = v0(x) ∈ TxS. Thereby v(x, t) = (v1, v2) ∈ TS denotes the tangential surface velocity, p(x, t) ∈ R the surface pressure, Re the surface Reynolds number, κ the Gaussian curvature, Tx§ the tangent space on x ∈ dR S, TS = ∪s∈STxS the tangent bundle and ∇v, ∇S· and ∆S the covariant directional derivative, surface divergence and surface Laplace-DeRham operator, respectively. As in flat space the equation results from conservation of mass and (tangential) linear momentum. However, differences are found in the appearing operators and the additional term including the Gaussian curvature. The unusual sign results from the definition of the Laplave-DeRham operator. While a huge literature exists for the two- dimensional Navier-Stokes equation in flat space, results for its surface counterpart are rare. We introduce a surface finite element approach which is also esirable for surfaces with genus g(S 6= 0, as it allows to deal with harmonic vector fields and demonstrate the dependency of the solution on the topology of the surface on various examples.

1 TU Dresden, Mathematics, Dresden, [email protected] 2 TU Dresden, [email protected] 30th Chemnitz FEM Symposium 2017 65

A Local Mesh Modiﬁcation Strategy for Interface Problems with Application to Shape and Topology Optimization

Peter Gangl1 Ulrich Langer2

In topology and shape optimization one is often interested in finding the optimal layout of a subdomain of a fixed computational domain. Most algorithms start out from an initial design and use sensitivity information of the objective functional with respect to the geometry in order to successively update the material interface to reach an optimum. Examples for such sensitivities are the topological derivative or the shape derivative. In PDE-constrained design optimization problems, these sensitivities usually depend on the solutions to the state equation and the adjoint equation, which have to be determined in each iteration of the optimization algorithm. When these two partial differential equations are solved by a standard finite element method, it is important that the material interface is resolved by the finite element mesh in order to obtain accurate approximate solutions. We present a local mesh modification strategy which adapts mesh nodes only in a vicinity of the material interface in such a way that, on the one hand, the interface is resolved accurately, and on the other hand, no angle can come too close to 180 degrees. This maximum angle condition allows to show optimal order of convergence of the finite element method independently of the location of the interface relative to the mesh. While the occurrence of too large angles is excluded by the procedure, angles can become arbitrarily small which affects the condition of the arising linear systems. This approach is an adaptation of the approach of Frei and Richter, where finite element methods on quadrilateral meshes are used, to the case of piecewise linear, globally continuous finite elements on triangular grids. We integrate this interface finite element method into a two-stage design optimization algorithm where the optimal topology is found by means of a level set algorithm that is based on the topological derivative before using shape optimization as a post- processing. We apply the presented optimization strategy to the design optimization of an electric motor.

References:

[1] S. Frei and T. Richter. A locally modiﬁed parametric ﬁnite element method for interface problems. SIAM J. Numer. Anal., 52(5):2315–2334, 2014.

1 TU Graz, Institut fur¨ Numerische Mathematik, Graz, Austria, [email protected] 2 Institute of Computational Mathematics, JKU Linz, [email protected] 66 30th Chemnitz FEM Symposium 2017

Finite Elements for Scalar Convection-Dominated Equations and Incompressible Flow Problems – a Never Ending Story

Volker John1 Petr Knobloch2 Julia Novo3

The contents of this paper is twofold. First, important recent results concerning ﬁnite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important problems in these ﬁelds are discussed.

1 Weierstrass Institute for Applied Analysis and Stochastics, Research Group Numerical Mathematics and Scientiﬁc Computing, Berlin, Germany, [email protected] 2 Department of Numerical Mathematics, Faculty of Mathematics and, Physics, Charles University, Sokolovska´ 83, 18675 Praha 8, Czech Republic, [email protected] 3 Departamento de Matematicas,´ Universidad Autonoma´ de Madrid, Spain, [email protected] 30th Chemnitz FEM Symposium 2017 67

Towards Pressure-robust Mixed Methods for the Incompressible Navier-Stokes Equations

Alexander Linke1

For more than thirty years it was thought that the efficient construction of pressure- robust mixed methods for the incompressible Navier–Stokes equations, whose velocity error is pressure-independent, was practically impossible. However, a novel, quite uni- versal construction approach shows that it is indeed rather easy to construct pressure- robust mixed methods. The approach repairs a certain L2-orthogonality between gradient fields and discretely divergence-free test functions, and works for families of arbitrary-order mixed finite element methods, arbitrary-order discontinuous Galerkin methods, and finite volume methods. Novel benchmarks for the incompressible Navier–Stokes equations show that the approach promises significant speedups in computational practice, when- ever the continuous pressure is complicated.

1 Weierstrass Institute, Numerical Mathematics and Scientiﬁc Computing, Berlin, Germany, [email protected] 68 30th Chemnitz FEM Symposium 2017

Pressure Robust Discretizations for Incompressible Flows

Philip Lukas Lederer1 Joachim Schoberl¨ 2

Classical inf-sup stable mixed finite elements for the incompressible (Navier–)Stokes equations are not pressure-robust, i.e., their velocity errors depend on the continuous pressure. However, a modification only in the right hand side of a Stokes discretization is able to reestablish pressure-robustness, as shown recently for several inf-sup stable Stokes elements. For the modification of the right hand side a velocity reconstruction operator is constructed that maps discretely divergence-free test functions to exactly divergence-free ones. This can be done with a simple element wise BDM interpolator in the case of discontinuous pressure approximations. Recently, this concept was extended to low and high order Taylor–Hood and mini elements, which have continuous discrete pressures and also for a new discretization with a relaxed H(div)-conformity. We present the basic concept and show different numerical examples to confirm that the new pressure-robust elements converge with optimal order and outperform significantly the classical versions of those elements.

References:

[1] Alexander Linke On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime Comput. Methods Appl. Mech. Engrg., Vol. 268, 782 – 800, 2014. [2] Linke, Alexander and Matthies, Gunar and Tobiska, Lutz, Robust arbitrary order mixed ﬁnite element methods for the incompressible Stokes equations with pressure independent velocity errors ESAIM Math. Model. Numer. Anal., Vol. 50, 289 – 309, 2016. [3] Lederer, Philip L. and Linke, Alexander and Merdon, Christian and Schoberl,¨ Joachim, Diver- gence-free Reconstruction Operators for Pressure-Robust Stokes Discretizations with Continu- ous Pressure Finite Elements SIAM Journal on Numerical Analysis, Vol. 55, 1291 – 1314, 2017. [4] Lederer, Philip L. and Christoph Lehrenfeld and Schoberl,¨ Joachim, [5] Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows. Part I arXiv:1707.02782, 2017.

1 TU Wien, Institute for Analysis an Scientiﬁc Computing, Vienna, Austria, [email protected] 2 Institute for Analysis an Scientiﬁc Computing - TU Wien, [email protected] 30th Chemnitz FEM Symposium 2017 69

Superconvergent Graded Meshes

Thomas Apel1 Mariano Mateos2 Johannes Pfefferer3 Arnd Rosch¨ 4

Superconvergent discretization error estimates can be obtained when the solution is smooth enough and the ﬁnite element meshes enjoy some structural properties. The simplest one is that any two adjacent triangles form a parallelogram. The solution of elliptic boundary value problems contains singularities in the vicinity of corners (and edges in 3D) leading to reduced convergence order in the case of quasiuniform meshes. A remedy is the use of graded meshes near these corners. The talk summarizes our results about a combination of both approaches.

1 Universitat¨ der Bundeswehr Munchen,¨ Institut fur¨ Mathematik und Bauinformatik, Neubiberg, Germany, [email protected] 2 Universidad de Oviedo, [email protected] 3 TU Munchen,¨ [email protected] 4 Universtat¨ Duisburg-Essen, [email protected] 70 30th Chemnitz FEM Symposium 2017

Functional A Posteriori Error Estimates for the Nonlinear Poisson-Boltzmann Equation

Svetoslav Nakov1 Johannes Kraus2 Sergey Repin3

In this talk, we show a short derivation of the Poisson-Boltzmann equation (PBE) and then the focus goes on deriving a functional a posteriori error estimate for the PBE. The advantage of the functional a posteriori error estimates based on the duality theory is that only the structure of the equation alone is exploited and therefore no global or local constants enter in the estimate. This is in contrast to other methods, e.g a residual based one, which depend on the particular triangulation. Therefore functional type a posteriori error estimates give not only an error indicator, but also a guaranteed bound on the error. We also show some numerical experiments.

1 RICAM, Computational Methods for PDEs, Linz, Austria, [email protected] 2 University of Duisburg-Essen , Faculty of Mathematics , [email protected] 3 V.A. Steklov Institute of Mathematics at St. Petersburg, [email protected] 30th Chemnitz FEM Symposium 2017 71

On the Stability and Conditioning of Anisotropic Finite-Element-Runge-Kutta Methods

Jens Lang1 Weizhang Huang2 Lennard Kamenski3

In [HKL2010,HKL2013] anisotropic mesh adaptation methods for elliptic problems are studied. In a next step, we have investigated the influence of anisotropic meshes upon the time stepping and the conditioning of the linear systems arising from linear ﬁnite element approximations of linear parabolic equations. Here, we present stability results and estimates for the condition number. Both explicit and implicit time integration schemes are considered. For stabilized explicit Runge-Kutta methods, it is shown that the allowed maximal step size depends only on the number of the elements in the mesh and a measure of the non-uniformity of the mesh viewed in the metric speci- ﬁed by the inverse of the diffusion matrix. Particularly, it is independent of the mesh non-uniformity in volume measured in the Euclidean metric [HKL2016]. For the implicit time stepping situation, bounds are established for the condition number of the resulting linear system with and without diagonal preconditioning for the implicit Euler (the simplest implicit RK method) and general implicit RK methods. It is shown that the conditioning of an implicit RK method behaves like that of the implicit Euler method. The obtained bounds for the condition number have explicit geometric interpretations and take the interplay between the diffusion matrix and the mesh geometry into full consideration. They show that there are three mesh-dependent factors that can affect the conditioning: the number of elements, the mesh non-uniformity measured in the Eu- clidean metric, and the mesh non-uniformity with respect to the inverse of the diffusion matrix. They also reveal that the preconditioning using the diagonal of the system matrix, the mass matrix, or the lumped mass matrix can effectively eliminate the effects of the mesh non-uniformity measured in the Euclidean metric [HKL2017]. Illustrative numerical examples are given.

References:

[1] W. Huang, L. Kamenski, J. Lang (HKL2010), A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates, J. Comp. Phys. 229 (2010), pp. 2179 2198. [2] W. Huang, L. Kamenski, J. Lang (HKL2013), Adaptive finite elements with anisotropic meshes, Numerical Mathematics and Advanced Applications 2011: Proceedings of ENUMATH 2011, the 9th European Conference on Numerical Mathematics and Advanced Applications, Leices- ter, September 2011, A. Cangiani et al. (eds.), pp. 33-42, Springer 2013. [3] W. Huang, L. Kamenski, J. Lang (HKL2016), Stability of explicit one-step methods for P1-finite element approximation of linear diffusion equations on anisotropic meshes, SIAM J. Numer. Anal. 54 (2016), pp. 1612-1634. [4] W. Huang, L. Kamenski, J. Lang (HKL2017), Conditioning of implicit Runge-Kutta integration for finite element approximation of linear diffusion equations on anisotropic meshes, arXiv:1703.06463.

1 Technische Universitaet Darmstadt, Mathematics, Darmstadt, Germany, [email protected] 2 Kansas University, [email protected] 3 WIAS Berlin, [email protected] 72 30th Chemnitz FEM Symposium 2017

An Optimal Order DG Time Discretization Scheme for Parabolic Problems with Non-homogeneous Constraints Igor Voulis1 Arnold Reusken2

We consider parabolic problems with non-homogeneous constraints. Standard problems of this kind include the heat equation with a non-homogeneous Dirchlet boundary condition and the following Stokes problem with an non-homogeneous divergence condition and a non-homogeneous boundary condition (in Ω × [0,T ], Ω ⊂ Rd): u0 − ∆u + ∇p = f divu = g

u|∂Ω = h

u(0) = u0. This problem can be seen as a parabolic problem in (an affine coset of) the space of divergence free functions with a Lagrange multiplier p and two non-homogeneous conditions: divu = g and u|∂Ω = h. If one applies standard DG in time sub-optimal results are obtained (cf. Table below). We present an analysis which explains the cause of this sub-optimal behavior. Based on this analysis we introduce a modification which leads to an optimal convergence order, non only for the energy norm of u, but also for the L2 norm of the Lagrange multiplier p. Furthermore, an optimal nodal superconvergence result for u is obtained. Our theoretical results are confirmed by numerical results. In the table below one can see that the temporal convergence order for the Lagrange multiplier is 1 for the standard method (SM) and 2 for our modified method (MM). In this experiment we used a P2 − P1 Taylor-Hood pair in space and linear functions in time. For the modified method we see that the spatial error dominates after a few temporal refinements (NT ).

NT 4 8 16 32 64 128 SM 1.36231 0.73455 0.37695 0.19011 0.09513 0.04755 EOCT 0.89112 0.96248 0.98752 0.99894 1.00042 MM 0.30828 0.07813 0.01984 0.00589 0.00286 0.00261 EOCT 1.98035 1.97707 1.75344 1.04000 0.13472 Error in L2-norm between exact p and the solution of the discrete problem.

References:

[1] V. Thomee, Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics). Springer-Verlag New York, Inc., 2006. [2] S. Hussain, F. Schieweck, and S. Turek, “A note on accurate and efﬁcient higher order Galerkin time stepping schemes for the nonstationary Stokes equations,” The Open Numerical Methods Journal, vol. 4, pp. 35–45, 2012. [3] D. Schotzau¨ and C. Schwab, “Time Discretization of Parabolic Problems by the hp-Version of the Dis- continuous Galerkin Finite Element Method,” SIAM Journal on Numerical Analysis, vol. 38, no. 3, pp. 837–875, 2000.

1 RWTH-Aachen University, Institut fuer Geometrie und Praktische Mathematik, Aachen, Germany, [email protected] 2 Institut fuer Geometrie und Praktische Mathematik, RWTH Aachen, [email protected] 30th Chemnitz FEM Symposium 2017 73

Relating FEM to FVM for Interface Problems in CFD

Susanne Hollbacher¨ 1

We consider multiphase flow problems such as particle-fluid and gas-fluid mixtures to derive suitable finite element spaces for the Navier-Stokes equations. There is still ongoing research on the suitable choice of discrete finite element spaces for velocity and pressure in order to acquire a stable finite element scheme. Furthermore, it is common strategy in multiphase flow applications that the emerging immersed interface is not resolved by the Eulerian grid. Instead, the forces on the immersed interface get included into the discrete spaces.

In a previous work on a stable discretisation for particle-fluid flows, see [2], [3], we derived suitable finite element spaces for the description of the interface forces between fluid and particles. One essential ingredient is the comparison to an according finite volume scheme [1]. Since FVM comprise discretised surface integrals they turned out to be convenient to capture the interaction forces on the immersed interface. The insights and ideas gained from that simplified model system was extended to the general case of fluid-fluid two phase flows. Within that talk we propose a finite element space which offers a new approach for the inclusion of forces arising on immersed interfaces. We will derive the distinguished properties of the defined shape functions and emphasize their positive impact on the numerical properties of the discrete scheme. As a proof of concept first numerical results will be presented. Beside the application to multiphase flow the proposed spaces give rise to a new approach for the construction of inf-sup stable finite element spaces for velocity and pressure: pressure-like forces naturally arise due to the new degrees of freedom. The end of the talk gives a short outlook to that promising direction.

References:

[1] Bank, R. E. and Rose, D. J.: Some Error Estimates for the Box Method. SIAM Journal on Nu- merical Analysis 22:777–787 (1987). [2] Hoellbacher, S.: Test space modeling for interface problems: A stable FV and FE scheme for the DNS of particulate flow. Part I: Rotational test spaces. (submitted) [3] Hoellbacher, S.: Test space modeling for interface problems: Projected Discrete Delta Func- tions with Application to the DNS of particulate flow. Part II: Flat-Top test spaces. (submitted)

1 KAUST, CEMSE, ECRC, Thuwal, Saudi-Arabien, [email protected] 74 30th Chemnitz FEM Symposium 2017

A Coupled FEM-FVM Method for Electroosmotic Flow

Jurgen¨ Fuhrmann1 Christian Merdon2 Alexander Linke3

Microscale electroosmotic flows occur in many interesting applications, including pore scale processes in fuel cell membranes and sensing with nanopores. We present a new approach for the numerical solution of coupled fluid flow and ion transport in a self-consistent electric ﬁeld. Ingredients of the method are

- Pressure-robust, pointwise divergence free ﬁnite element discretization of the Stokes equations describing the barycentric velocity of the ionic mixture

- Thermodynamically consistent, maximum principle observing ﬁnite volume method for ion transport including competition for ﬁnite available volume

- Coupling approach between fluid flow and mass transport together with a ﬁxed point iteration to solve the combined system.

The talk introduces the discretization approach and provides ﬁrst results of numerical simulations conﬁrming the validity of the approach. A number of open problems and challengig directions will be described.

References:

[1] A. Linke. On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Computer methods in applied mechanics and engineering, 268:782–800, 2014. [2] Volker John, Alexander Linke, Christian Merdon, Michael Neilan, and Leo G Rebholz. On the divergence constraint in mixed ﬁnite element methods for incompressible flows. SIAM Review, 2017. WIAS Preprint 2177, to appear. [3] J. Fuhrmann. Comparison and numerical treatment of generalised Nernst–Planck models. Computer Physics Communications, 196:166–178, 2015. [4] J. Fuhrmann. A numerical strategy for Nernst–Planck systems with solvation effect. Fuel cells, 16(6):704–714, 2016. [5] Ch. Merdon, J. Fuhrmann, A. Linke, F. Neumann, T. Streckenbach, H. Baltruschat, and M. Kho- dayari. Inverse modeling of thin layer flow cells for detection of solubility, transport and reaction coefﬁcients from experimental data. Electrochimca Acta, pages 1–10, 2016.

1 Weierstrass Institute, Numerical Mathematics and Scientific Computing, Berlin, [email protected] 2 Weierstrass Institute, Numerical Mathematics and Scientific Computing, Berlin, [email protected] 3 Weierstrass Institute, Numerical Mathematics and Scientific Computing, Berlin, [email protected] 30th Chemnitz FEM Symposium 2017 75

Fluid-Structure Interaction with H(div)-Conforming HDG and a new H(curl)-Conforming Method for Non-Linear Elasticity

Michael Neunteufel1 Joachim Schoberl¨ 2

Fluid-structure-interaction problems arise in a variety of engineering applications and finding appropriate discretization is still challenging. Often Taylor-Hood elements for the fluid and H1-conforming elements for the solid are used, as they are easy to implement, however they entail some disadvantages. In this talk we present a new kind of coupling of the Navier-Stokes equations with the elastic wave equation using mixed methods. The H(div)-conforming Hybrid Discontinuous Galerkin method is used for the discretization of the Navier-Stokes equations, which brings a new term in the Arbitrary Lagrangian Eulerian description besides the appearing mesh velocity. For the elasticity part we introduce a new method, which is based on the idea to use H(curl)-conforming elements for the velocity instead of standard H1-elements. There- fore an additional variable is needed: the impulse, for which we use the dual space of H(curl). The method is implemented in NGS-Py, which is based on the finite element library Netgen/NGSolve (www.ngsolve.org). Finally, we present first numerical results.

1 Vienna University of Technology, Institute for Analysis and Scientiﬁc Computing, Vienna, Austria, [email protected] 2 Vienna University of Technology, Institute for Analysis and Scientiﬁc Computing, [email protected] 76 30th Chemnitz FEM Symposium 2017

A new Approach for Kirchhoff-Love Plates and Shells

Walter Zulehner1 Katharina Rafetseder2

A new approach is introduced for deriving a mixed variational formulation for Kirch- hoff plate bending problems with mixed boundary conditions involving clamped, simply supported, and free boundary parts. Based on a regular decomposition of an appropriate nonstandard Sobolev space for the bending moments, the fourth-order problem can be equivalently written as a system of three (consecutively to solve) second-order problems in standard Sobolev spaces. This leads to the design of new discretization methods, which are flexible in the sense, that any existing and well-working discretization method and solution strategy for standard second-order problems can be used as a modular building block of the new method. Essential features of this approach can be extended to Kirchhoff-Love shells.

1 Johannes Kepler University Linz, Institute of Computational Mathematiccs, Linz, Austria, [email protected] 2 Johannes Kepler University Linz, [email protected] 30th Chemnitz FEM Symposium 2017 77

hp-FEM for a Stabilized Three-ﬁeld Formulation of the Biharmonic Problem

Jan Petsche1 Lothar Banz2 Andreas Schroder¨ 3

In this talk, we present a stabilized three-field formulation of the biharmonic problem ∆2u = f. The need for a discrete inf-sup-condition for the resulting saddle point problem is circumvented by least-squares-like consistent stabilization terms. A priori error estimates for appropriate norms are derived and a reliable and efficient residual error estimator based on an implicit H2-reconstruction is shown. Several numerical examples confirm the applicability of the proposed techniques.

1 University of Salzburg, Department of Mathematics, Salzburg, Austria, [email protected] 2 University of Salzburg, Department of Mathematics, Salzburg, Austria, [email protected] 3 University of Salzburg, Department of Mathematics, Salzburg, Austria, [email protected] 78 30th Chemnitz FEM Symposium 2017

Efﬁcient Simulation of Short Fibre Reinforced Composites

Rolf Springer1 Arnd Meyer2

Lightweight structures became more and more important over the last years. One special class of such structures are short ﬁbre reinforced composites, produced by in- jection moulding. To avoid expensive experiments for testing the mechanical behaviour of these composites proper material models are needed. Thereby, the stochastic nature of the ﬁbre orientation is the main problem. In this talk we look onto the simulation of such materials in a linear thermoelastic setting. So, we use the stress-strain relation

σ = C :(ε − (θ − θ0)T), with a fourth order material tensor C, a second order thermal expansion tensor T, the temperature difference (θ − θ0), and the second order linearised strain tensor ε. The temperature ﬁeld θ within this equation is described by

−∇ · (κ · ∇θ) = Θ, whereas θ0 describes a reference ﬁeld. In the last equation κ describes the heat con- duction and is a symmetric second order tensor. In both equations the material properties (κ, T, and C) depend on the stochastic ﬁbre orientation. Thereby, the classical approach is to average these quantities and solve the above equations with the averaged expression. We will present a way how this approach can be extended to achieve better approximations of the solutions. For this setting we will present some numerical results.

1 TU Chemnitz, Mathematics, Chemnitz, Germany, [email protected] 2 TU Chemnitz, [email protected] 30th Chemnitz FEM Symposium 2017 79

Automated Finite Element Assembling

Matthias Hochsteger1 Joachim Schoberl¨ 2

In this talk we present implementation aspects of the general purpose Finite Ele- ment software NGSolve. In particular we address the steps to transform a variational formulation given by the user in a high-level representation into an algorithm to assem- ble element matrices. We also discuss the differences between run-time evaluation and just-in-time compilation where C++ code is generated at run-time for the given equation.

1 TU Wien, Analysis and Scientiﬁc Computing, Wien, Austria, [email protected] 2 TU Wien, [email protected] 80 30th Chemnitz FEM Symposium 2017

How to Make a Common Lisp Finite Element Library High-performing?

Nicolas Neuß1 Marco Heisig2

We describe the optimization and parallelization of the Finite Element library FEM- LISP using a 3D elasticity problem as a model example. This model problem features some special difﬁculties, but also several advantages which can be exploited to achieve a satisfactory compromise between high-level flexibility and low-level performance.

1 FAU Erlangen-Nurnberg,¨ Mathematik, Erlangen, Germany, [email protected] 2 FAU Erlangen-Nurnberg,¨ Lehrstuhl Informatik 10, [email protected] 30th Chemnitz FEM Symposium 2017 81

Domain Decomposition and Memory Footprint Reduction of an Eikonal Solver

Daniel Ganellari1 Gundolf Haase2 Gerhard Zumbusch3

The basis equations in cardiac electrophysiology are the bidomain equations describing the intercellular and the extracellular electrical potential via a system of two PDEs coupled nonlinearly by a bunch of ODEs. Its difference, the transmembrane potential, is responsible for the excitation of the heart and its steepest gradients form an excitation wavefront propagating in time. This arrival time ϕ(x) of the wavefront at some point x ∈ Ω can be approximated by the Eikonal equation [1] q (∇ϕ(x))T M(x)∇ϕ(x) = 1 x ∈ Ω with given heterogeneous, anisotropic velocity information M. The domain Ω ⊂ R3 is discretized by planar-sided tetrahedrons with a piecewise linear approximation of the solution ϕ(x) inside each of them. The numerical solution of the Eikonal equation fol- lows the fast iterative method [2] with its application for tetrahedral meshes [3]. Therein the main operations in each discretization element τ contain various inner products in T τ the M-metric as h~ek,s,~es,`iM τ ≡ ~ek,s · M · ~es,` with ~es,` as connecting edge between vertices s and ` in element τ. While the authors of [3] pass all coordinates of the tetrahedron together with the 6 entries of M τ we precompute these inner products and use only them in the wave front computation. This ﬁrst change requires less memory trans- fers for each tetrahedron.

The second change is caused by the fact that h~ek,s,~es,`iM τ (k 6= `) represents an angle of a surface triangle whereas h~ek,s,~ek,siM τ represents the length of an edge in the M-metric. Basic geometry as well as vector arithmetics yield to the conclusion that the angle information can be expressed by the combination of three edge lengths. There- fore we only have to precompute the 6 edge lengths of a tetrahedron and compute the remaining 12 angle data on-the-fly which reduces the memory footprint per tetrahedron to 6 numbers. The efﬁcient implementation of the two changes requires a local Gray-code number- ing of edges in the tetrahedron and a bunch of bit shifts to assign the appropriate data. Numerical experiments on CPUs and GPUs show that the reduced memory footprint approach is faster by 40% than the original implementation. Additionally, we will present our very recent domain decomposition algorithm for the Eikonal equation. For large scale problems, the task based parallel model will run into

1 Karl Franzens University of Graz, Institute for Mathematics and Scientific Computing, 8010, Graz, Austria, [email protected] 2 Karl Franzens University of Graz, Institute for Mathematics and Scientific Computing, 8010 , Graz, , Austria, [email protected] 3 Friedrich-Schiller-Universitat¨ Jena, Institut fur¨ Angewandte Mathematik, 07743 Jena, Germany, [email protected] 82 30th Chemnitz FEM Symposium 2017 difficulties: There might be not enough (shared) memory on a single host or on a GPU, the computing power of a single compute unit is not sufficient, or the parallel efficiency is not satisfactory. In all cases, a distributed memory model is needed. Hence a coarser decomposition of the algorithm is needed, namely a domain decomposition approach. The domain Ω is statically partitioned into a number of non-overlapping sub-domains Ωi. Each of them is assigned to a single processor. Synchronization and communication of the processors is to be reduced to a minimum. In our case, a single processor i can efficiently solve the Eikonal equation on Ωi, as long as its boundary data on ∂Ωi is correct. However, this data may belong to the outer boundary ∂Ω or to other processors. Hence inter-processor communication is needed. We present two different strategies on load balancing in CUDA in order to achieve to run the domain decomposition approach in one GPU. The first approach maps simply one sub-domain to one thread block. Its scaling improves with an increased number of sub-domains by reducing the overall host synchronization together with the preallocation of the global memory. The second approach takes better advantage of the GPU shared memory since it shares the workload of one sub-domain between many thread blocks exploiting in this way the total shared memory space. This allows to overcome the shared memory limitation with sufficient sub-domains which improves the performance significantly. This works very well if enough GPU memory is available. Otherwise we have to preallocate data for each block in each iteration which drops the performance significantly with increased number of sub-domains. This GPU memory limitations can be relaxed by allowing memory allocations only by the active blocks computing for one active sub-domain in the wave front. Again this preallocation is performed in each iteration but only from those blocks who are currently run on one SM. As soon as the blocks finish their execution the memory is freed and ready to be used by other active blocks waiting to be distributed on the idle SMs. The domain decomposition approach is the first step towards the inter-process communication implementation where the limitation of the global memory will be overcome completely by using multiple accelerator cards and cluster computing. As a future work, it will allow the preallocation of global memory which will enable the scalability on large scale problems. Supported in part by the FWF project F32-N18 and the JoinEU-SEE scholarship.

References:

[1] J. P. Keener, An Eikonal-curvature equation for action potential propagation in myocardium, J. Math. Biol., 29 (1991), pp. 629-651. [2] W.-K. Jeong and R. T. Whitaker, A fast iterative method for eikonal equations, SIAM J. Sci. Comput., 30(5), pp. 2512-2534, 2008. [3] Z. Fu , R. M. Kirby and R. T. Whitaker, A fast iterative method for solving the eikonal equation on tetrahedra domains, Sci. Comput. 35(5), pp. C473-C494, 2013. List of Participants 84 30th Chemnitz FEM Symposium 2017 [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] from e-mail , Marius Paul [ 44 ] Hamburg Germany ¨ urgen [ 74 ] Berlin Germany , Matthias [ 79 ] Wien Austria , Susanne [ 73 ] Thuwal Saudi Arabia , Bernhard [ 27 ] Linz Austria ,J , Clemens [ 56 ] Linz Austria , Mark [ 50 ] Providence RI USA , Nicole [ 41 ] Bilbao Spain , Daniel [ 30 ] Linz Austria , Helmut [ 58 ] Basel Switzerland , Alex [ 19 ] Birmingham United Kingdom , first name Abstr. , Daniel [ 81 ] Graz Austria , Sven [ 15 ] Bonn Germany , Andrej Siegen Germany , Andreas [ 45 ] Erlangen Germany , Roland [ 47 ] Chemnitz Germany ¨ auser , Simon [ 37 ] Dresden Germany , Gabriel [ 43 ] Buenos Aires Argentina , Chaobao [ 52 ] Beijing China , Armin Linz Austria , Maksim [ 36 ] St. Petersburg Russia , Gundolf [ 13 ] Graz Austria , Peter [ 65 ] Graz Austria , Herbert [ 20 ] Darmstadt Germany , Christoph [ 29 , 48 ] Linz Austria , Paolo [ 42 ] Aachen Germany , Christoph [ 28 , 46 ] Darmstadt Germany , Lothar [ 18 ] Salzburg Austria , Stefan [ 62 ] Graz Austria , Thomas [ 69 ] Neubiberg Germany ¨ ollbacher Surname Acosta Ainsworth Apel Banz Becher Bespalov Beuchler Brenner Bruchh Cusimano Dohr Egger Endtmayer Erath Frolov Fohler Fuhrmann Ganellari Gangl Garanza Gatto Haase Harbrecht Herzog Hochsteger Hofer Hofreither Huang H Jodlbauer 30th Chemnitz FEM Symposium 2017 85 [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] ¨ ucken Germany from e-mail , Svetlana [ 39 ] Linz Austria , Michael [ 75 ] Vienna Austria , Oliver [ 25 ] Freiberg Germany , Martin [ 60 ] Linz Austria , Svetozar [ 32 ] Sofia Bulgaria , Sergej [ 57 ] Saarbr , Ariel [ 22 ] Rosario Argentina , Ricardo H. [ 26 ] College Park MD USA , Andreas [ 17 ] Salzburg Austria , first name Abstr. , Johannes [ 33 ] Garching Germany , Vadim [ 35 ] St. Petersburg Russia , Jan [ 77 ] Salzburg Austria , Philip Lukas [ 68 ] Vienna Austria , Ewald Linz Austria , Jens Markus [ 34 ] Wien Austria , Ulrich Linz Austria , Eglantina [ 31 ] Tirana Albania , Felix [ 55 ] Linz Austria , Bert Linz Austria , Matthias Chemnitz Germany , Svetoslav [ 70 ] Linz Austria , Arnd Chemnitz Germany , Francisco-Javier [ 21 ] Newark DE USA ¨ uller , Alexander [ 67 ] Berlin Germany , Nicolas [ 80 ] Erlangen Germany , Johanna Graz Austria , Michael Dresden Germany , Volker [ 66 ] Berlin Germany , Jens [ 71 ] Darmstadt Germany ¨ unther [ 59 ] Graz Austria ¨ oder , Mario [ 38 ] Linz Austria ,G ¨ uttler Schr Scholz Sayas Rjasanow Petsche Pfefferer Rheinbach Of Pester Nakov Neum Neunteufel Neuß Nochetto Meyer J Kalluci Kapl Korneev Lang Langer Lederer Lindner Linke Lombardi Margenov Matculevich Mayr Melenk Jung Surname John 86 30th Chemnitz FEM Symposium 2017 [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] from e-mail , Ioannis [ 40 ] Linz Austria , Olaf [ 53 ] Graz Austria , first name Abstr. , Joachim [ 14 ] Vienna Austria , Walter [ 76 ] Linz Austria , Rolf [ 78 ] Chemnitz Germany , Lutz [ 63 ] Magdeburg Germany , Zoa [ 24 ] Oxford United Kingdom , Gabriel [ 12 ] Thuwal Saudi Arabia , Stefan [ 49 ] Linz Austria , Martin [ 51 ] Beijing China , Igor [ 72 ] Aachen Germany , Annette Linz Austria , Michael [ 16 ] Chemnitz Germany , Axel [ 64 ] Dresden Germany , Huidong [ 61 ] Graz Austria , Marco [ 54 ] Graz Austria ¨ oberl Springer Steinbach Stynes Takacs Surname Sch Tobiska Toulopoulos Voigt Voulis Weihs Weise de Wijn Wittum Yang Zank Zulehner

88 Additional Organisational Hints

Internet access The BifEB offers free internet access (WLAN).

Food The conference fee includes:

• tea, coffee, soft drinks and snacks during breaks,

• the conference dinner on Tuesday.

Main meals are included in the full board price of the BifEB. If you haven’t booked a room with full board, you should buy vouchers for the meals at BifEB’s reception before 9:00 am.

Recreation The BifEB offers ﬁtness room, sauna, games room and more for free (→ “Lindenhaus”).

Conference Dinner The conference dinner takes place on Tuesday at 7:30 pm in the “Haupthaus” of BifEB.

Excursion The excursion will take place on Tuesday. – Extra costs for tickets: 45 Euro. We will walk through Strobl to the boat which leaves at 1:20 pm. Therefore, we will meet at 1 pm in front of the BifEB. In St. Wolfgang we take the cog railway (Schafbergbahn) at 2:50 pm to go uphill to the summit station (Schafbergspitze) arriving at 3:25 pm. Return: The last boat from St. Wolfgang leaves at 5:05 pm and reaches Strobl at 5:40 pm, thus you would have to leave the Schafberg via railway after a stay of 1 hour at 4:25 pm. Alternatively, you can leave the Schafberg with the last train at 5:05 pm, then it is a walking distance of about 1 hour from the valley station in St. Wolfgang to Strobl. BifEB Location Plan 89 www.tu-chemnitz.de/mathematik/fem-symposium/

Fakultat¨ fur¨ Mathematik www.tu-chemnitz.de/mathematik/

Technische Universitat¨ Chemnitz 09107 Chemnitz www.tu-chemnitz.de